Parameters of directed strongly regular graphs

Sylvia A. Hobart & aeb, 010105.

We give parameters, constructions and nonexistence information for directed strongly regular graphs as defined by Duval [9].

Definition
Hadamard matrices
The 2-dimensional case
The 1-dimensional case
Combinatorial parameter conditions
No Abelian Cayley graphs
No prime number of vertices
Constructions and Nonexistence Conditions
Uniqueness and enumeration
References


Table 1-20
Table 21-30
Table 31-40
Table 41-50
Table 51-60
Table 61-70
Table 71-80
Table 81-90
Table 91-100
Table 101-110

Definition

A directed strongly regular graph (dsrg) is a (0,1) matrix A with 0's on the diagonal such that the linear span of I, A and J is closed under matrix multiplication. This concept was defined by Duval [9], and most, if not all, of the theory given below can be found there. Earlier, Hammersley [20] studied the "love problem", one version of which is a special case of the problem considered here.

One defines (integral) parameters v, k, t, λ, μ by: A is a matrix of order v, and AJ = JA = kJ, and A2 = tI + λA + μ(J–I–A).

In case A is symmetric, we have a strongly regular graph, and that case is excluded here. This means that we require t < k.

In particular, we exclude the case A = J–I, so that the algebra spanned by I, A and J is 3-dimensional. It follows that A has precisely 3 distinct eigenvalues, say k, r, s, with multiplicities 1, f and g, respectively.

These dsrg's come in complementary pairs: together with A also J–I–A satisfies the definition. If the first one has parameters v, k, t, λ, μ then its complement has parameters v, v–k–1, v–2k–1+t, v–2k–2+μ, v–2k+λ. In the tables below we give complementary pairs together.

For a given set of parameters, dsrg's also come in pairs: together with A also its transpose satisfies the definition.

Hadamard matrices

We shall exclude one other case, namely that where t = 0. The theory is as follows.

First of all, the eigenvalues r, s different from k are roots of x2 + (μ–λ)x + μ–t = 0 so are algebraic integers. Next, if their multiplicities f, g differ then we can solve r, s from r+s = λ–μ and f.r + g.s = –k to find that r and s are rational numbers, and therefore integers. At least one is negative since A has trace 0. But J–I–A has eigenvalues v–1–k, –1–s, –1–r, and also has a negative eigenvalue, so we may assume that r is nonnegative and s is negative. In particular, r and s differ, some linear combination of A, I and J is a projection, and A is diagonalizable. Moreover, r.s is nonpositive, so μ ≤ t. In particular, t is nonzero.

Remains the case where f = g = (v–1)/2. From (λ–μ)(v–1)/2 = –k it follows that k = (v–1)/2 and μ = λ+1. From k2 = t + λ.k + μ(v–1–k), i.e., k2 = t + k(2μ–1), it follows that k divides t. But we already excluded t = k, so t = 0 and k = 2μ–1. The graph is a tournament: the transpose of A is J–I–A, and bordering the (1,–1) matrix J–2A first with a first column of all –1's and then with a top row of all 1's we find a Hadamard matrix H of order 4μ that has 1's on the diagonal and is skew symmetric off-diagonal. Conversely, any such H gives rise to such A.

The 2-dimensional case

An important subclass is that where already the linear span of A and J is closed under matrix multiplication. This happens when t = μ, and then A2 = (λ–μ)A + μ.J so that the eigenvalues of A are k, λ–μ, and 0. Conversely, when A has eigenvalue 0, we are in this case. Put d=μ–λ. Then the multiplicities of the eigenvalues k, –d, 0 are 1, k/d and v–1–k/d respectively.

The 1-dimensional case

If A is a 0-1 matrix, then B = J – 2A is a ±1 matrix, and it is possible that the 1-space generated by B is closed under matrix multiplication. This happens when t = μ and v–4k+4t = 2d (with d as in above - clearly this is a subcase of the 2-dimensional case). Conversely, if B is a ±1 matrix with constant row sums and 1's on the diagonal such that B2 is a multiple of B, then A = (J–B)/2 is the adjacency matrix of a dsrg in this subcase. Note that the set of such ±1 matrices B is closed under taking tensor products.

Combinatorial parameter conditions

We already saw that 0 < μ ≤ t. (If μ = 0 then A = J–I, which was excluded.) Also, that 0 ≤ λ < t < k < v. (Indeed, λ < t, since λ+1–t = (r+1)(s+1) ≤ 0.)

Duval gave one more condition, and parameter sets violating it will not be mentioned in the table. We have –2(k–t–1) ≤ μ–λ ≤ 2(k–t). (Indeed, consider a directed edge from x to y. Paths of length 2 from x to y contribute to λ, paths in the opposite direction to μ. Thus, the difference between μ and λ is counted by the at most 2(k–t) paths that cannot be reversed. The other inequality follows similarly, or by applying the first to the complementary graph.)

No Abelian Cayley graphs

From the spectrum we can draw one more useful conclusion. (Klin et al. [29]) Let us write A = As + Aa where As is the symmetric 0-1 matrix (with row sums t) describing adjacency via an undirected edge, and Aa is the 0-1 matrix describing the remaining, directed, edges (with row sums k–t). Since Aa has a nonzero real eigenvalue, namely k–t, and its square has trace zero, Aa must also have non-real eigenvalues. On the other hand, both A and As only have real eigenvalues. It follows that A and As cannot be diagonalised simultaneously, so that As and Aa do not commute. But then these matrices do not describe differences in the same Abelian group. Thus, a directed strongly regular graph cannot be a Cayley graph of an Abelian group.

No prime number of vertices

Duval [9] showed that the parameter conditions imply that v is not prime.

Constructions and Nonexistence Conditions

T1

For every k > 1, there exists a dsrg with v = k(k+1), k = k, t = 1, μ = 1, λ = 0. (Duval [9]) This is the special case of T8(i) obtained by taking a 2-(k+1,2,1) design.

T2

If k is even, there exists a dsrg with v = k2 – 1, k = k, t = 2, μ = 1, λ = 1. (Duval [9]) This is the special case μ = 1 of T4.

T3

Let q be a prime power congruent 1 (mod 4). Then there exists a dsrg with v = 2q, k = q–1, t = μ = λ+1 = k/2. (Duval [9])

T4

Let μ, k be positive integers such that μ divides k–1. Then there exists a dsrg with v = (k+1)(k–1)/μ, t = μ+1, λ = μ (Jørgensen [24]). Jørgensen's construction is as follows: take as vertices the integers mod v and let x→y be an edge when x+ky = 1,2,...,k (mod v).

T5

Let n be an odd integer. Then there exists a dsrg with v = 2n, k = n–1, t = μ = λ+1 = k/2. (Klin et al. [29])

T6

Let n be an even integer. Then there exists a Cayley dsrg with v = 2n, k = n–1, t = μ+1 = λ+1 = n/2. (Hobart & Shaw [21]) These parameters are a special case of those found under T4.

T7

If there exists a generalized quadrangle GQ(a,b), then there exists a dsrg with v = (a+1)(b+1)(ab+1), k = ab+a+b, t = λ+1 = a+b, μ = 1. (Klin et al. [30]) This graph is obtained by looking at (point,line) flags, where (p,L)→(q,M) when the flags are distinct and q is on L.

T8

If there exists a 2-(V,B,R,K,Λ) design, then (i) there exists a dsrg with v = VR, k = R(K–1), t = μ = Λ(K–1), λ = Λ(K–2) and also (ii) a dsrg with v = VR, k = RK–1, t = λ+1 = Λ(K–1)+R–1, μ = ΛK. (Fiedler et al. [12]) The former is obtained by looking at (point,block) flags, where (p,B)→(q,C) when p and q are distinct and q is on B. The latter is obtained by looking at (point,block) flags, where (p,B)→(q,C) when the flags are distinct and q is on B.

T9

If there exists a partial geometry with parameters pg(K,R,T) then there exists a dsrg with v = RK(1 + (K–1)(R–1)/T), k = RK–1, t = λ+1 = K+R–2, μ = T. (Fiedler et al. [12]) This graph is obtained by looking at (point,line) flags, where (p,L)→(q,M) when the flags are distinct and q is on L. Of course T7 is the special case T = 1 of this.

T10

If there exists a dsrg with parameters v, k, t, λ, μ, and t = μ, then for each positive integer m there also is a dsrg with parameters mv, mk, mt, m*λ, m*μ. (Duval [9]) Construction: given A, consider A ⊗ J, with J of order m.

T11

If there exists a dsrg with parameters v, k, t, λ, μ, and t = λ+1, then for each positive integer m there also is a dsrg with parameters mv, m(k+1)–1, m(t+1)–1, m(λ+2)–2, mμ. (Duval [9]) Construction: given A, consider (A+I) ⊗ J – I. In other words: apply T10 to the complements.

T12

Suppose n, d, and c are positive integers such that c < n–1 and c divides d(n–1). Then there exists a dsrg with parameters v = dn(n–1)/c, k = d(n–1), t = μ = c.d, λ= (c–1)d. (Godsil et al. [14]) Construction: take the complete directed graph on n vertices, each edge with multiplicity d. Colour the edges with d(n–1)/c colours in such a way that all in- and outdegrees for each fixed colour are c. Let the vertices of the dsrg be the arrow bundles consisting of all arrows of a given colour starting at a given vertex. Make a directed edge from one arrow bundle to another when one of the edges of the first one ends at the vertex of the second one.

T13

There do exist dsrg with parameters (v,k,t,λ,μ) = (24,8,3,2,3), (24,9,7,2,4), (24,10,8,4,4), (26,11,7,4,5). (Jørgensen [26])

T14

Let m and q be positive integers, and assume that each prime power in the complete factorization of m is 1 mod q. Then there exists a Cayley dsrg with v = m.q, k = m–1, t = μ = λ+1 = (m–1)/q. (Duval & Iourinski [10]) These parameters are a special case of those found under T12.

T15

Let q be an odd prime power. Then there exists a dsrg with v = (q–1)(q2–1), k = q2–2, t = λ+1 = 2q–2, μ = q. (Fiedler et al. [11])

T16

Let q be an odd prime power. There there exists a dsrg with v = 2q2, k = 3q–2, t = 2q–1, λ = q–1, μ = 3. (Fiedler et al. [11])

T17

Let q be a prime power. There there exists a dsrg with v = hq2(q–1), k = hq(q–1), t = μ = hq–h+1, λ = (h–1)(q–1), where 0 < h ≤ q. (Olmez & Song [36]) Construction: take the antiflags (p,L) in a net consisting of h parallel classes of lines in AG(2,q), and let pL→rM when p is on M.

More generally, consider a 1½-design with point-block incidence matrix N (cf. Neumaier [35]) and parameters r, k, a, b, so that NJ = rJ and JN = kJ and NNtN = aN + bJ. Then the directed graph with as vertices the flags of this design and with adjacency pB → qC when the flags are distinct and p is in C is a directed strongly regular graph, and NNtN = (λ+2)N + μ(J – N). These graphs have t = λ+1. Similarly, the directed graph with as vertices the antiflags of this design, with the same adjacency, is a directed strongly regular graph, and N(J–N)tN = λN + μ(J – N). These graphs have t = μ.

T18

There exist dsrg with v = r(1+ab)2b, k = r(1+ab)ab, t = μ = ra2b+a, λ = ra2b+a–ab–1, where r, a, b are positive integers with r ≥ 2.

There also exist dsrg with v = (m+1)s, k = ls, t = μ = ld, λ = ld–d, where d, l, m, s are positive integers with dm = ls, and 1 ≤ l < m.

There also exist dsrg with v = (m+1)s, k = ls+s–1, t = ld+s–1, λ = ld+s–2, μ = ld+d, where d, l, m, s are positive integers with dm = ls, and 1 ≤ l < m. (Olmez & Song [37])

T19

Let q be a prime power, and let s ≤ q+1. Then there exists a dsrg with parameters v = sq2, k = sq–1, t = q+s–2, λ = q+s–3, μ = s–1.

Let q be a prime power, and let s ≤ q. Then there exists a dsrg with parameters v = sq2, k = (s+1)q–2, t = 2q+s–3, λ = q+s–3, μ = s+1. (Araluze et al. [1]. Note that the authors forgot the condition s ≤ q+1 in their Proposition 3.4; some of the examples they mention are incorrect.)

T20

(Gyürki [15]) Suppose there exists a dsrg with parameters v, k, t, λ, μ, and a partition C1, C2, ..., Ca of its vertex set into a parts of size b each, such that for each pair g, h the number of arcs from Cg to a fixed vertex x in Ch is independent of the choice of x, and is equal to μ if g ≠ h and to λ+b–k if g = h. Then for any positive integer u, there exists a dsrg with parameters (ua+1)v, uv+k, ub+t, ub+λ, ub+μ. That is, with parameters u(av, v, b, b, b) + (v, k, t, λ, μ).

Construction: Let Γ be the given dsrg, with vertex set V. The new graph Γ' has vertex set V × {0, 1, ..., ua}. There is an arc (x,g) → (y,g) if there is an arc x → y in Γ; there is an arc (x,g) → (y,h), where g ≠ h, when y is in Ci and h is in g + (i–1)u + {1,...,u} (mod ua+1).

If Γ has eigenvalues k, r, s, then the eigenvalues of Γ' are uv+k, r, s.

For example, if Γ is the Hoffman-Singleton graph with parameters 50, 7, 7, 0, 1 and the partition is one into five Petersen graphs (so that a=5, b=10), we find dsrg Γ' with parameters 250u+50, 50u+7, 10u+7, 10u, 10u+1 for any positive integer u.

T21

(Zhou, He & Chai [38])

T22

Crnković & Švob [6] construct a dsrg(63,11,8,1,2) invariant under PSL(2,8), where the undirected part is an antipodal 7-cover of the complete graph K9.

T23

Byzov & Pushkarev [3] construct a dsrg(22,9,6,3,4) invariant under a cyclic group of order 3.

T24

Van Hulst [23] constructs a dsrg(k(k+1)/t, k, t, t–1, t) invariant under a cyclic group of order k for (k,t) = (9,2), (11,3), (11,2), (13,2), (17,6). Also a dsrg(40,10,3,1,3) invariant under 24:2 of order 32, and a dsrg(40,15,6,5,6) invariant under a cyclic group of order 8, and a dsrg(35,14,6,5,6) invariant under a cyclic group of order 7.

T25

Byzov & Pushkarev [4] construct a dsrg(9(2n+3),3(2n+3),2n+4,2n+1,2n+4) for all n ≥ 1.

T26

Gyürki & Klin (unpub. ms, [19]) construct dsrg(2n2, 3n–2, 2n–1, n–1, 3) for all n ≥ 2. (For n=6 these parameters are new. For odd prime powers n, see also T16.) Also, a dsrg(2n2, 4n–2, 2n+2, n+2, 6) for all n ≥ 3. (For n=6 these parameters are new.) Also, a dsrg(3n2, 4n–2, 2n, n, 4) for all n ≥ 2. (For prime powers n ≥ 3, see also T19.) Also a dsrg(3n2, 6n–2, 2n+6, n+6, 10) for all n ≥ 3.

M1

If there exists a srg with parameters v, k, λ, μ, and μ = λ+1, then there is a dsrg with parameters vk, (v–k–1)k, (v–k–1)(k–μ), (v–k–2)(k–μ), (v–k–1)(k–μ). Construction: take the edges of the srg, and let xy→uv when u is at distance 2 from y. Example: the Petersen graph produces a dsrg(30,18,12,10,12).

M2

If there exists a srg with parameters v, k, λ, μ, and μ = λ, then there is a dsrg with parameters vk, (v–k)k, (v–k)(k–μ), (v–k–1)(k–μ), (v–k)(k–μ). Construction: take the edges of the srg, and let xy→uv when u is at distance 0 or 2 from y. Example: from srg(16,6,2,2) we get a dsrg(96,60,40,36,40).

M3

There exists a dsrg(18,4,3,0,1). Construction: Consider AG(2,3) with a fixed direction, say vertical. Let the vertices of the dsrg be the 9 points and the 9 non-vertical lines. Take an undirected edge (pair of opposite directed edges) for each incident (point,line) pair. Let g be an automorphism of order 3 moving points vertically, and add directed edges x→gx and gL→L. (Cf. Duval [9], Theorem 4.4.)

M4

Let r be an odd prime power. Then there exists a dsrg with parameters 2r2, (r–1)(2r+1)/2, (r–1)(3r+1)/4, r(r–1)/2–1, r(r–1)/2. Construction: Take two copies of the finite field of order r2. Join vertices in one copy when they differ by a square, in the other when they differ by a non-square. Thus, both halves are undirected (Paley) graphs. View the field as AG(2,r) and pick two sets, B and C, each the union of (r–1)/2 mutually parallel lines, where B and C use different directions. Make an arrow from x in the first copy to y' in the second copy when y–x is a member of B, and an arrow from y' to x when y–x is a member of C. Example: we get dsrg(18,7,5,3,2) and dsrg(50,22,16,10,9).

M5

There exist precisely 1292 nonisomorphic dsrg(15,5,2,1,2), namely 1174, 100, 10, 5, 3 with full automorphism group of order 1, 2, 3, 4, 5, respectively, according to Jørgensen [25]. (Is there also one with a simple description?)

M6

There exist precisely 16495 nonisomorphic dsrg(14,6,3,2,3), see below. Maybe the first was given by Duval [9].

M7

The tensor product of two ±1 matrices belonging to dsrgs in the 1-dimensional case, is again a dsrg in the 1-dimensional case. Here one may also take I of order 2.

M8

(Godsil, Hobart & Martin [14]) Given a dsrg with adjacency matrix A, such that μ=λ and v=4k–4μ, and a Hadamard matrix H of order c with constant row and column sums d, and ones on the diagonal, put B = J – 2A and B' = B ⊗ H. The parameter conditions say that B is a ±1 matrix with 1's on the diagonal, constant row and column sums, and with B2 a multiple of I. Clearly, the same holds for B'. Hence A', given by B' = J – 2A', is again the adjacency matrix of a dsrg. The parameters are v' = vc, k' = 2c(k–μ)+d(2μ–k), t' = c(k+t–2μ)+d(2μ–k), λ' = μ' = c(k–μ)+d(2μ–k). Note that c = d2.

M9

(Martinez & Araluze [32]) Using partial sum families in a cyclic group, the authors construct examples for (v,k,t,λ,μ) = (27,10,6,3,4), (28,7,2,1,2), (30,13,8,5,6), (32,9,6,1,3), (32,10,7,3,3), (33,11,4,3,4), (34,14,12,5,6), (34,15,9,6,7), (35,8,4,1,2), (39,10,6,1,3), (40,17,11,6,8), (42,14,5,4,5), (44,10,4,3,2), (45,14,6,5,4).

M10

(Gyürki & Klin [18]) Using unions of classes in an association scheme, the authors construct examples for (v,k,t,λ,μ) = (30,13,11,6,5), (32,9,6,1,3), (32,13,9,4,6), (32,14,10,6,6), (36,13,7,4,5), (36,13,11,2,6), (39,10,6,1,3), (39,12,4,3,4), (39,14,6,5,5), (39,16,12,7,6), (45,16,8,5,6), (48,10,6,2,2), (48,13,7,2,4), (50,16,10,3,6), (50,23,13,10,11), (54,8,3,2,1), (54,16,12,6,4), (54,19,9,6,7), (54,20,16,6,8), (54,21,17,8,8), (54,25,14,11,12), (60,13,5,2,3), (60,26,20,10,12), (63,22,10,7,8), (72,19,11,2,6), (72,20,14,4,6), (72,21,15,6,6), (72,22,9,6,7), (72,26,18,8,10), (84,29,19,6,12), (84,31,17,12,11), (84,39,27,18,18), (90,28,16,10,8), (105,36,16,11,13). (Gyürki wrote that the (72,26,10,8,10) in the paper should have been (72,26,18,8,10).)

M11

(Jørgensen [28]) There exist dsrgs with parameters (36,10,5,2,3), (96,13,5,0,2), (108,10,3,0,1) related to mixed Moore graphs.

M12

(Martinez [33]) Using partial sum families one can construct dsrgs with parameters (18,5,3,2,1), (18,7,5,2,3), (27,8,4,3,2), (27,10,6,3,4), (32,7,4,3,1), (32,9,6,1,3), (36,11,5,4,3), (36,13,7,4,5), (39,16,12,7,6), (45,14,6,5,4), (45,16,8,5,6), (48,11,5,4,2), (50,9,5,4,1), (64,15,6,5,3), (75,14,6,5,2).

M13

(Michel [34]) From products of groups one gets several infinite families of parameters that cover the small examples (80,22,10,6,6), (81,28,12,9,10), (99,34,14,11,12).

M14

(Crnković, Švob & Žutolija [7]) There exist dsrgs with parameters (60,21,11,6,8), (60,22,12,8,8), (60,24,10,9,10), (60,25,17,8,12), (60,27,21,12,12) and (60,28,20,14,12) invariant under the group S5 × 2.

M15

(aeb, unpub) There exist dsrgs with parameters (72,28,18,12,10) invariant under S5 × S4, with parameters (90,26,16,6,8) invariant under S5 × S3, with parameters (90,32,20,10,12) invariant under S6, and with parameters (60,18,11,6,5) invariant under S5.

M16

(Crnković, Švob & Žutolija [8]) There exist dsrgs with parameters (108,11,3,2,1), (108,14,10,0,2), (108,22,12,6,4), (108,23,9,8,4), (108,25,15,8,5), (108,34,18,12,10), (108,38,22,12,14), (108,39,23,14,14), (108,41,35,16,15), (108,42,33,18,15) and (108,46,22,19,20). And dsrgs with parameters (54,10,4,1,2), (54,11,4,3,2), (54,16,7,4,5).

M17

AI work by Stephen Chung & William J. Wesley [5]: existence for 143 new parameter sets.

N1

There is no dsrg with v = 14, k = 5, t = 4, λ = 1, μ = 2. (Klin et al. [29])

N2

There is no dsrg with v = 32, k = 6, t = 5, λ = 1, μ = 1. (Jørgensen [24])

N3

If t = μ and g = 2 then k divides v. (Indeed, all rows of A are linear combinations of two rows and the all-one vector. If k does not divide v then the all-one vector cannot be used, unless k = 2v/3 = t, an excluded case.)

N3a

More generally, Jørgensen [26] shows that if g = 2 then we have either (v,k,t,λ,μ) = (6m,2m,m,0,m) or (v,k,t,λ,μ) = (8m,4m,3m,m,3m).

N4

(Jørgensen [26]) If g = 3 then we have either (v,k,t,λ,μ) = (6m,3m,2m,m,2m) or (v,k,t,λ,μ) = (12m,3m,m,0,m).

N5

(Jørgensen [26]) We have (k–t)(μ–k+t) ≤ λ.t. (Indeed, fix a point x and look at triangles xyz where xy is a directed edge and xz is an undirected edge, and yz may be directed (in the yz direction) or undirected. There are at least the LHS and at most the RHS such triangles.)

N6

(Jørgensen [26]) If λ = 0 and μ = k–t, then v is a multiple of 3μ. (Indeed, since λ = 0, the μ triangles on a directed edge contain directed edges only, and one sees that the directed edges form a graph of which each connected component has 3μ vertices (since it is the μ-coclique extension of a directed triangle).)

N7

Jørgensen [26] showed that dsrg's satisfy the usual absolute bound:

(i) If s ≠ –1 then v ≤ f(f+3)/2. If moreover v(r+s2) ≠ 2(r–s)(k–s) then even v ≤ f(f+1)/2.

(ii) if r ≠ 0 then v ≤ g(g+3)/2. If moreover v(s+r2) ≠ 2(s–r)(k–r) then even v ≤ g(g+1)/2.

N8

There is no dsrg with v = 30, k = 7, t = 5, λ = 0, μ = 2. (Jørgensen [26])

N9

(Jørgensen [27]) In the 2-dimensional case, if A has rank 5, so that t = μ and g = 4 (i.e., k = 4(μ–λ)), then the parameter set (v,k,t,λ,μ) is one of (20m,4m,m,0,m), (36m,12m,5m,2m,5m), (10m,4m,2m,m,2m), (16m,8m,5m,3m,5m), (20m,12m,9m,6m,9m), (18m,12m,10m,7m,10m).

N10

(Hobart & Williford [22]) If a dsrg has an independent set of size c, then c ≤ f, and if r ≠ 0, then c ≤ g.

In particular, if λ=0 then k ≤ f, and if also r ≠ 0, then k ≤ g.

In particular, if λ=1 then k/3 ≤ f, and if also r ≠ 0, then k/3 ≤ g.

N11

(Hobart & Williford [22]) Consider the graph induced on the outneighbours of a fixed vertex x. If k > f (or k > g) then it has eigenvalue s (resp. r) with geometric multiplicity at least k–f (resp. k-g).

It follows that if k > f (or k > g) then λ ≥ –s (resp. λ ≥ r).

N12

Dsrgs with λ=0 and μ=1 are known as mixed Moore graphs. They have v = k2 + k + 1 – t. López, Miret & Fernández [31] show the nonexistence of such graphs for the cases (k,t) = (6,3), (7,4), (9,7) by computer.

N13

In [5] (Paper 5) it is shown that if g=5 and t=μ then v/k takes one of the values 3/2, 2, 3, 4, 6. This rules out 19 parameter sets in our table. For g=6 there are partial results that rule out 2 parameter sets with v=56.

N14

Studying graphs with k=t+1 (Paper 31) eliminates the parameter set (30,12,11,4,5). Extending this work (Paper 55) also eliminates (86,38,37,16,17).

Uniqueness and enumeration

The two smallest dsrg's (with parameters (6,2,1,0,1) and (8,4,3,1,3)) are unique, as one easily checks. Matrices are for example 

001100
000011
000011
110000
110000
001100
and 
00001111
00001111
01010101
10101010
01010101
10101010
11110000
11110000

(Uniqueness of dsrg(8,3,2,1,1), the complement of the one above, was first shown in Hammersley [20].)

But in general there is large freedom. Here is a dsrg with parameters (10,4,2,1,2) where points come in pairs with equal in-neighbours, but there is no such pairing for out-neighbours. 

0000001111
0011110000
1100110000
0000001111
0000001111
1111000000
1100000011
0011110000
0011110000
1100001100

Jørgensen [25] did an exhaustive search and finds 16 dsrg(10,4,2,1,2).

Hammersley [20] found a dsrg(15,4,2,1,1) and is inclined to believe that it is the only one. However, Jørgensen [25] determined that there are precisely 5 of these.

Table of Jørgensen's enumeration results:

v k t λ μ # comment
6 2 1 0 1 1
8 3 2 1 1 1 Hammersley [20]
10 4 2 1 2 16
12 3 1 0 1 1
12 4 2 0 2 1
12 5 3 2 2 20
14 5 4 1 2 0 Klin et al. [29]
14 6 3 2 3 16495
15 4 2 1 1 5
15 5 2 1 2 1292
16 6 3 1 3 0 Fiedler et al. [11]
16 7 5 4 2 1
18 4 3 0 1 1
18 5 3 2 1 2
18 6 3 0 3 1
20 4 1 0 1 1
24 6 2 0 2 1

More generally, Gimbert [13] showed that dsrg(k(k+1),k,1,0,1) is unique. Jørgensen [26] shows more generally that dsrg(m(m+1)t,mt,t,0,t) is unique for all positive integers m and t. And also that dsrg(8m,4m,3m,m,3m) is unique for all positive integers m.

Small vertex transitive graphs

Fiedler, Klin & Muzychuk [11] found all vertex transitive dsrg on at most 20 vertices. Gyürki and Brouwer [17], [2] extended this and found all vertex transitive dsrg on at most 39 vertices. Gyürki [16] gives pictures and a detailed description of the dsrg on at most 12 vertices.

Below a table. The columns headed #, #C, #D give entries for the number (up to isomorphism) of all vertex transitive dsrgs with k < v/2, the number of such Cayley graphs, and the number of such Cayley graphs for the dihedral group. The entries are pairs a/s, where a ("all") is the total count, and s ("symmetric") is the count of such graphs isomorphic to their transpose (obtained by reversing all arrows).

Note that v cannot be prime, and v must be even for #D. The number of graphs up to isomorphism and symmetry is (a+s)/2.

v k t λ μ # #C #D
6 2 1 0 1 1/1 1/1 1/1
8 3 2 1 1 1/1 1/1 1/1
10 4 2 1 2 3/1 1/1 1/1
12 3 1 0 1 1/1 1/1 0/0
12 4 2 0 2 1/1 1/1 1/1
12 5 3 2 2 4/2 4/2 2/2
14 6 3 2 3 3/1 3/1 3/1
16 7 4 3 3 3/1 3/1 1/1
16 7 5 4 2 1/1 1/1 1/1
18 4 3 0 1 1/1 1/1 0/0
18 5 3 2 1 2/0 2/0 0/0
18 6 3 0 3 1/1 1/1 1/1
18 7 5 2 3 4/0 2/0 0/0
18 8 4 3 4 8/2 5/1 5/1
18 8 5 4 3 1/1 1/1 1/1
20 4 1 0 1 1/1 1/1 0/0
20 7 4 3 2 2/0 2/0 0/0
20 8 4 2 4 5/1 5/1 1/1
20 9 5 4 4 8/2 6/2 2/2
21 6 2 1 2 4/0 4/0 -
21 8 4 3 3 3/1 3/1 -
22 10 5 4 5 7/1 7/1 7/1
24 5 2 1 1 6/0 4/0 0/0
24 6 2 0 2 1/1 1/1 0/0
24 7 3 2 2 10/2 10/2 0/0
24 8 3 2 3 8/0 8/0 0/0
24 8 4 0 4 1/1 1/1 1/1
24 9 7 2 4 2/0 2/0 0/0
24 10 8 4 4 6/0 6/0 0/0
24 11 6 5 5 17/1 11/1 3/1
24 11 7 6 4 6/2 6/2 2/2
24 11 8 7 3 1/1 1/1 1/1
26 11 7 4 5 4/0 0/0 0/0
26 12 6 5 6 16/2 12/2 12/2
27 8 4 3 2 3/1 3/1 -
27 10 6 3 4 7/1 7/1 -
28 12 6 4 6 6/2 3/1 3/1
28 13 7 6 6 6/2 6/2 6/2

v k t λ μ # #C #D
30 5 1 0 1 1/1 0/0 0/0
30 9 3 2 3 4/0 0/0 0/0
30 9 5 4 2 2/0 0/0 0/0
30 10 4 2 4 2/0 0/0 0/0
30 10 5 0 5 1/1 1/1 1/1
30 11 5 4 4 3/1 0/0 0/0
30 12 6 3 6 5/1 1/1 1/1
30 13 8 5 6 4/0 0/0 0/0
30 13 11 6 5 2/0 0/0 0/0
30 14 7 6 7 41/1 31/1 31/1
30 14 8 7 6 5/1 1/1 1/1
30 14 9 8 5 1/1 1/1 1/1
32 7 4 3 1 2/0 2/0 0/0
32 9 6 1 3 12/0 8/0 0/0
32 10 7 3 3 22/0 18/0 0/0
32 13 9 4 6 40/6 36/6 0/0
32 14 10 6 6 85/15 73/15 0/0
32 15 8 7 7 7/1 7/1 3/1
32 15 9 8 6 8/2 8/2 1/1
32 15 11 10 4 1/1 1/1 1/1
34 14 12 5 6 2/0 0/0 0/0
34 15 9 6 7 4/0 0/0 0/0
34 16 8 7 8 41/1 33/1 33/1
36 8 2 1 2 4/0 2/0 0/0
36 9 3 0 3 1/1 1/1 0/0
36 10 5 2 3 14/0 14/0 0/0
36 11 5 4 3 12/2 7/1 0/0
36 11 7 6 2 2/0 2/0 0/0
36 12 5 2 5 2/0 2/0 0/0
36 12 6 0 6 1/1 1/1 1/1
36 13 7 4 5 59/1 50/0 0/0
36 13 11 2 6 1/1 1/1 0/0
36 16 8 6 8 20/4 17/3 5/1
36 17 9 8 8 47/7 33/5 10/2
36 17 11 10 6 4/2 4/2 2/2
38 18 9 8 9 63/1 63/1 63/1
39 10 6 1 3 1/1 1/1 -
39 12 4 3 4 17/1 17/1 -
39 14 6 5 5 16/0 16/0 -
39 16 12 7 6 4/0 4/0 -

v k t λ μ # #C #D
40 8 2 0 2 1/0 1/0 0/0
40 9 3 2 2 9/1 9/1 0/0
40 15 9 8 4 2/0 2/0 0/0
40 16 8 4 8 5/1 5/1 1/1
40 19 10 9 9 41/1 13/1 7/1
40 19 11 10 8 2/2
40 19 14 13 5 1/1 1/1 1/1
42 6 1 0 1 1/1 1/1 0/0
42 11 6 5 2 2/0 2/0 0/0
42 12 4 2 4 8/0 8/0 0/0

Totals:

v # #C #D
6 1/1 1/1 1/1
8 1/1 1/1 1/1
10 3/1 1/1 1/1
12 6/4 6/4 3/3
14 3/1 3/1 3/1
16 4/2 4/2 2/2
18 17/5 12/4 7/3
20 16/4 14/4 3/3
21 7/1 7/1 -
22 7/1 7/1 7/1
24 58/8 50/8 7/5
26 20/2 12/2 12/2
27 10/2 10/2 -
28 12/4 9/3 9/3
30 71/7 35/5 35/5
32 177/25 153/25 5/3
34 47/1 33/1 33/1
36 167/19 134/14 18/6
38 63/1 63/1 63/1
39 38/2 38/2 -

(And none of orders 9, 15, 25, 33, 35.)

The 728 vertex-transitive dsrgs with v < 40, k < v/2 are given explicitly (in GAP/GRAPE format) in verttra_dsrg_list.

Table 1-20

v k t λ µ rf sg comments
6 2 1 0 1 03 −12T1
T5
T8(i) for 2-(3,2,1)
T12
T18 for (d,l,s)=(1,1,2)
3 2 1 2 02 −13T8 (ii) for 2-(3,2,1)
T9 for pg(2,2,2)
T18 for (d,l,s)=(1,1,2)
T20
8 3 2 1 1 12 −15T4
T6
T7 for GQ(1,1)
T19
4 3 1 3 05 −22T17
T18 for (a,b,r)=(1,1,2)
T19
10 4 2 1 2 05 −14T3
T5
T12
T18 for (d,l,s)=(1,2,2)
M1 from srg(5,2,0)
5 3 2 3 04 −15T18 for (d,l,s)=(1,2,2)
T20
12 3 1 0 1 08 −13T1
T8(i) for 2-(4,2,1)
T12
T18 for (d,l,s)=(1,1,3)
8 6 5 6 03 −18T8 (ii) for 2-(4,3,2)
T18 for (d,l,s)=(2,2,3)
12 4 2 0 2 09 −22T8(i) for 2-(3,2,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,1,4)
7 5 4 4 12 −19T8 (ii) for 2-(3,2,2)
T11 for m=2
T18 for (d,l,s)=(2,1,4)
12 5 3 2 2 13 −18T4
T6
T8 (ii) for 2-(4,2,1)
T9 for pg(3,2,2)
T11 for m=2
T18 for (d,l,s)=(1,1,3)
T19
6 4 2 4 08 −23T8(i) for 2-(4,3,2)
T10 for m=2
T12
T18 for (a,b,r)=(1,1,3)
T18 for (d,l,s)=(2,2,3)
14 5 4 1 2 17 −26does not exist by N1
8 7 4 5 16 −27
14 6 3 2 3 07 −16T5
T12
T18 for (d,l,s)=(1,3,2)
M6
7 4 3 4 06 −17T18 for (d,l,s)=(1,3,2)
T20
15 4 2 1 1 15 −19T2
T4
10 8 6 8 09 −25
15 5 2 1 2 09 −15M5
9 6 5 6 05 −19
16 6 3 1 3 012 −23does not exist by N4
9 6 5 5 13 −112
16 7 4 3 3 14 −111T4
T6
T15
8 5 3 5 011 −24T18 for (a,b,r)=(1,1,4)
16 7 5 4 2 32 −113T11 for m=2
8 6 2 6 013 −42T10 for m=2
18 4 3 0 1 110 −27M3
13 12 9 10 17 −210
18 5 3 2 1 24 −113T7 for GQ(1,2)
T19
M12
12 10 7 10 013 −34T18 for (a,b,r)=(2,1,2)
18 6 3 0 3 015 −32T8(i) for 2-(3,2,3)
T10 for m=3
T12
T17
T18 for (d,l,s)=(3,1,6)
11 8 7 6 22 −115T8 (ii) for 2-(3,2,3)
T11 for m=3
T18 for (d,l,s)=(3,1,6)
18 7 5 2 3 19 −28T16
T19
M4
M12
10 8 5 6 18 −29
18 8 4 3 4 09 −18T3
T5
T12
T18 for (d,l,s)=(1,4,2)
T20
9 5 4 5 08 −19T18 for (d,l,s)=(1,4,2)
T20
18 8 5 4 3 23 −114T11 for m=3
9 6 3 6 014 −33T10 for m=3
20 4 1 0 1 015 −14T1
T8(i) for 2-(5,2,1)
T12
T18 for (d,l,s)=(1,1,4)
15 12 11 12 04 −115T8 (ii) for 2-(5,4,3)
T18 for (d,l,s)=(3,3,4)
20 7 4 3 2 24 −115T8 (ii) for 2-(5,2,1)
T9 for pg(4,2,2)
T18 for (d,l,s)=(1,1,4)
12 9 6 9 015 −34T8(i) for 2-(5,4,3)
T12
T18 for (d,l,s)=(3,3,4)
20 8 4 2 4 015 −24T10 for m=2
T12
T18 for (d,l,s)=(2,2,4)
11 7 6 6 14 −115T11 for m=2
T18 for (d,l,s)=(2,2,4)
20 9 5 4 4 15 −114T4
T6
T11 for m=2
10 6 4 6 014 −25T10 for m=2
T18 for (a,b,r)=(1,1,5)

Table 21-30

v k t λ µ rf sg comments
21 6 2 1 2 014 −16T8(i) for 2-(7,3,1)
T12
T18 for (d,l,s)=(1,2,3)
14 10 9 10 06 −114T18 for (d,l,s)=(2,4,3)
21 8 4 3 3 16 −114T8 (ii) for 2-(7,3,1)
T9 for pg(3,3,3)
T18 for (d,l,s)=(1,2,3)
12 8 6 8 014 −26T12
T18 for (d,l,s)=(2,4,3)
22 9 6 3 4 111 −210T23
12 9 6 7 110 −211
22 10 5 4 5 011 −110T5
T12
T18 for (d,l,s)=(1,5,2)
11 6 5 6 010 −111T18 for (d,l,s)=(1,5,2)
T20
24 5 2 1 1 19 −114T4
18 15 13 15 014 −29
24 6 2 0 2 020 −23T8(i) for 2-(4,2,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,1,6)
17 13 12 12 13 −120T8 (ii) for 2-(4,3,4)
T11 for m=2
T18 for (d,l,s)=(4,2,6)
24 7 3 2 2 18 −115T4
T11 for m=2
16 12 10 12 015 −28T10 for m=2
24 8 3 2 3 015 −18T13
T20
15 10 9 10 08 −115
24 8 4 0 4 021 −42T8(i) for 2-(3,2,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,1,8)
15 11 10 8 32 −121T8 (ii) for 2-(3,2,4)
T11 for m=2
T18 for (d,l,s)=(4,1,8)
24 9 7 2 4 115 −38T13
14 12 8 8 28 −215
24 10 5 3 5 018 −25does not exist by N13
13 8 7 7 15 −118
24 10 8 4 4 29 −214T13
13 11 6 8 114 −39
24 11 6 5 5 16 −117T4
T6
12 7 5 7 017 −26T18 for (a,b,r)=(1,1,6)
24 11 7 6 4 33 −120T8 (ii) for 2-(4,2,2)
T11 for m=2
T18 for (d,l,s)=(2,1,6)
12 8 4 8 020 −43T8(i) for 2-(4,3,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,2,6)
24 11 8 7 3 52 −121T11 for m=3
12 9 3 9 021 −62T10 for m=3
25 9 6 5 2 43 −121
15 12 7 12 021 −53does not exist by N4
25 10 6 1 6 022 −52does not exist by N3
14 10 9 6 42 −122
26 11 7 4 5 113 −212T13
14 10 7 8 112 −213
26 12 6 5 6 013 −112T3
T5
T12
T18 for (d,l,s)=(1,6,2)
13 7 6 7 012 −113T18 for (d,l,s)=(1,6,2)
T20
27 7 4 1 2 115 −211?
19 16 13 14 111 −215?
27 8 4 3 2 26 −120T9 for pg(3,3,2)
T19
M12
18 14 11 14 020 −36T18 for (a,b,r)=(2,1,3)
27 9 4 1 4 023 −33does not exist by N4
17 12 11 10 23 −123
27 10 6 3 4 114 −212T19
M9
M12
16 12 9 10 112 −214
27 11 6 5 4 25 −121
15 10 7 10 021 −35does not exist by N13
27 12 8 2 8 024 −62does not exist by N3
14 10 9 5 52 −124
28 6 3 2 1 27 −120?
21 18 15 18 020 −37?
28 7 2 1 2 020 −17M9
20 15 14 15 07 −120
28 12 6 4 6 021 −26T8(i) for 2-(7,4,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,3,4)
15 9 8 8 16 −121T8 (ii) for 2-(7,4,2)
T11 for m=2
T18 for (d,l,s)=(2,3,4)
28 13 7 6 6 17 −120T4
T6
T11 for m=2
14 8 6 8 020 −27T10 for m=2
T18 for (a,b,r)=(1,1,7)
30 5 1 0 1 024 −15T1
T8(i) for 2-(6,2,1)
T12
T18 for (d,l,s)=(1,1,5)
24 20 19 20 05 −124T8 (ii) for 2-(6,5,4)
T18 for (d,l,s)=(4,4,5)
30 7 5 0 2 120 −39does not exist by N8
22 20 16 16 29 −220
30 9 3 2 3 020 −19T12
T18 for (d,l,s)=(1,3,3)
20 14 13 14 09 −120T18 for (d,l,s)=(2,6,3)
30 9 5 4 2 35 −124T8 (ii) for 2-(6,2,1)
T9 for pg(5,2,2)
T11 for m=2
T18 for (d,l,s)=(1,1,5)
20 16 12 16 024 −45T8(i) for 2-(6,5,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,4,5)
30 10 4 2 4 024 −25T8(i) for 2-(6,3,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,2,5)
19 13 12 12 15 −124T11 for m=2
T18 for (d,l,s)=(3,3,5)
30 10 5 0 5 027 −52T8(i) for 2-(3,2,5)
T10 for m=5
T12
T18 for (d,l,s)=(5,1,10)
19 14 13 10 42 −127T8 (ii) for 2-(3,2,5)
T11 for m=5
T18 for (d,l,s)=(5,1,10)
30 11 5 4 4 19 −120T11 for m=2
T18 for (d,l,s)=(1,3,3)
18 12 10 12 020 −29T10 for m=2
T12
T18 for (d,l,s)=(2,6,3)
M1 from srg(10,3,0)
30 11 9 2 5 121 −48?
18 16 11 10 38 −221?
30 12 6 3 6 025 −34T8(i) for 2-(5,3,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,2,6)
17 11 10 9 24 −125T8 (ii) for 2-(5,3,3)
T11 for m=3
T18 for (d,l,s)=(3,2,6)
30 12 11 4 5 215 −314does not exist by N14
17 16 9 10 214 −315
30 13 8 5 6 115 −214M9
16 11 8 9 114 −215
30 13 11 6 5 39 −220M10
16 14 7 10 120 −49
30 14 7 6 7 015 −114T5
T12
T18 for (d,l,s)=(1,7,2)
T20
15 8 7 8 014 −115T18 for (d,l,s)=(1,7,2)
T20
30 14 8 7 6 25 −124T8 (ii) for 2-(6,3,2)
T11 for m=3
T18 for (d,l,s)=(2,2,5)
15 9 6 9 024 −35T10 for m=3
T12
T18 for (d,l,s)=(3,3,5)
30 14 9 8 5 43 −126T11 for m=5
15 10 5 10 026 −53T10 for m=5

Table 31-40

v k t λ µ rf sg comments
32 6 5 1 1 214 −217does not exist by N2
25 24 19 21 117 −314
32 7 4 3 1 36 −125T7 for GQ(1,3)
T19
M12
24 21 17 21 025 −46T18 for (a,b,r)=(3,1,2)
32 9 6 1 3 121 −310M9
M10
M12
22 19 15 15 210 −221
32 10 7 3 3 213 −218T19
M9
21 18 13 15 118 −313
32 11 6 5 3 35 −126
20 15 11 15 026 −45does not exist by N13
32 12 6 2 6 028 −43does not exist by N4
19 13 12 10 33 −128
32 13 9 4 6 120 −311M10
18 14 10 10 211 −220M8
32 14 7 5 7 024 −27M17
17 10 9 9 17 −124
32 14 10 6 6 212 −219M8
M10
17 13 8 10 119 −312
32 15 8 7 7 18 −123T4
T6
16 9 7 9 023 −28T18 for (a,b,r)=(1,1,8)
32 15 9 8 6 34 −127T11 for m=2
16 10 6 10 027 −44T10 for m=2
32 15 11 10 4 72 −129T11 for m=2
16 12 4 12 029 −82T10 for m=2
33 10 4 3 3 111 −121T4
22 16 14 16 021 −211
33 11 4 3 4 021 −111M9
21 14 13 14 011 −121
33 12 8 3 5 121 −311?
20 16 12 12 211 −221?
34 14 12 5 6 217 −316M9
19 17 10 11 216 −317
34 15 9 6 7 117 −216M9
18 12 9 10 116 −217
34 16 8 7 8 017 −116T3
T5
T12
T18 for (d,l,s)=(1,8,2)
17 9 8 9 016 −117T18 for (d,l,s)=(1,8,2)
T20
35 6 2 1 1 114 −120T2
T4
28 24 22 24 020 −214
35 8 4 1 2 120 −214M9
26 22 19 20 114 −220
35 12 8 4 4 214 −220M17
22 18 13 15 120 −314
35 14 6 5 6 020 −114T24
20 12 11 12 014 −120
36 8 2 1 2 027 −18T8(i) for 2-(9,3,1)
T12
T18 for (d,l,s)=(1,2,4)
27 21 20 21 08 −127T18 for (d,l,s)=(3,6,4)
36 9 3 0 3 032 −33T8(i) for 2-(4,2,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,1,9)
26 20 19 18 23 −132T8 (ii) for 2-(4,3,6)
T11 for m=3
T18 for (d,l,s)=(6,2,9)
36 10 5 2 3 120 −215M11
25 20 17 18 115 −220
36 11 5 4 3 28 −127T8 (ii) for 2-(9,3,1)
T9 for pg(4,3,3)
T11 for m=3
T18 for (d,l,s)=(1,2,4)
T19
M12
24 18 15 18 027 −38T10 for m=3
T12
T18 for (a,b,r)=(2,1,4)
T18 for (d,l,s)=(3,6,4)
36 11 7 6 2 54 −131T11 for m=2
24 20 14 20 031 −64T10 for m=2
36 12 5 2 5 031 −34T17
T18 for (a,b,r)=(1,2,2)
23 16 15 14 24 −131
36 12 6 0 6 033 −62T8(i) for 2-(3,2,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,1,12)
23 17 16 12 52 −133T8 (ii) for 2-(3,2,6)
T11 for m=2
T18 for (d,l,s)=(6,1,12)
36 13 7 4 5 119 −216M10
M12
22 16 13 14 116 −219
36 13 11 2 6 127 −58M10
22 20 14 12 48 −227
36 14 7 6 5 27 −128?
21 14 11 14 028 −37?
36 16 8 6 8 027 −28T10 for m=2
T12
T18 for (d,l,s)=(2,4,4)
M1 from srg(9,4,1)
19 11 10 10 18 −127T11 for m=2
T18 for (d,l,s)=(2,4,4)
36 16 14 8 6 49 −226M17
19 17 8 12 126 −59
36 17 9 8 8 19 −126T4
T6
T11 for m=2
18 10 8 10 026 −29T10 for m=2
T18 for (a,b,r)=(1,1,9)
M7
36 17 11 10 6 53 −132T8 (ii) for 2-(4,2,3)
T11 for m=2
T18 for (d,l,s)=(3,1,9)
18 12 6 12 032 −63T8(i) for 2-(4,3,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,2,9)
M7
38 16 13 6 7 219 −318?
21 18 11 12 218 −319?
38 17 10 7 8 119 −218?
20 13 10 11 118 −219?
38 18 9 8 9 019 −118T5
T12
T18 for (d,l,s)=(1,9,2)
19 10 9 10 018 −119T18 for (d,l,s)=(1,9,2)
T20
39 10 6 1 3 126 −312M9
M10
28 24 20 20 212 −226
39 12 4 3 4 026 −112T12
T18 for (d,l,s)=(1,4,3)
M10
26 18 17 18 012 −126T18 for (d,l,s)=(2,8,3)
39 14 6 5 5 112 −126T18 for (d,l,s)=(1,4,3)
M10
24 16 14 16 026 −212T12
T18 for (d,l,s)=(2,8,3)
39 16 12 7 6 312 −226M10
M12
22 18 11 14 126 −412
40 6 3 0 1 124 −215does not exist by N12
33 30 27 28 115 −224
40 8 2 0 2 035 −24T8(i) for 2-(5,2,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,1,8)
31 25 24 24 14 −135T8 (ii) for 2-(5,4,6)
T11 for m=2
T18 for (d,l,s)=(6,3,8)
40 9 3 2 2 115 −124T4
T11 for m=2
30 24 22 24 024 −215T10 for m=2
40 9 6 5 1 55 −134
30 27 21 27 034 −65does not exist by N13
40 10 3 1 3 034 −25T24
29 22 21 21 15 −134
40 11 4 3 3 114 −125T20
28 21 19 21 025 −214
40 15 6 5 6 024 −115T24
24 15 14 15 015 −124
40 15 9 8 4 54 −135T8 (ii) for 2-(5,2,2)
T11 for m=2
T18 for (d,l,s)=(2,1,8)
24 18 12 18 035 −64T8(i) for 2-(5,4,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,3,8)
40 15 12 3 7 130 −59?
24 21 15 13 49 −230?
40 16 8 4 8 035 −44T10 for m=2
T12
T18 for (d,l,s)=(4,2,8)
23 15 14 12 34 −135T11 for m=2
T18 for (d,l,s)=(4,2,8)
40 17 11 6 8 125 −314M9
22 16 12 12 214 −225
40 18 9 7 9 030 −29M17
21 12 11 11 19 −130
40 18 12 8 8 215 −224M17
21 15 10 12 124 −315
40 18 15 9 7 410 −229M17
21 18 9 13 129 −510
40 19 10 9 9 110 −129T4
T6
20 11 9 11 029 −210T18 for (a,b,r)=(1,1,10)
40 19 11 10 8 35 −134T11 for m=2
20 12 8 12 034 −45T10 for m=2
40 19 14 13 5 92 −137T11 for m=5
20 15 5 15 037 −102T10 for m=5

Table 41-50

v k t λ µ rf sg comments
42 6 1 0 1 035 −16T1
T8(i) for 2-(7,2,1)
T12
T18 for (d,l,s)=(1,1,6)
35 30 29 30 06 −135T8 (ii) for 2-(7,6,5)
T18 for (d,l,s)=(5,5,6)
42 11 6 5 2 46 −135T8 (ii) for 2-(7,2,1)
T18 for (d,l,s)=(1,1,6)
30 25 20 25 035 −56T8(i) for 2-(7,6,5)
T12
T18 for (d,l,s)=(5,5,6)
42 12 4 2 4 035 −26T8(i) for 2-(7,3,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,2,6)
29 21 20 20 16 −135T11 for m=2
T18 for (d,l,s)=(4,4,6)
42 12 9 4 3 314 −227?
29 26 19 22 127 −414?
42 13 5 4 4 114 −127T4
T11 for m=2
28 20 18 20 027 −214T10 for m=2
42 14 5 4 5 027 −114T20
M9
27 18 17 18 014 −127
42 14 7 0 7 039 −72T8(i) for 2-(3,2,7)
T10 for m=7
T12
T18 for (d,l,s)=(7,1,14)
27 20 19 14 62 −139T8 (ii) for 2-(3,2,7)
T11 for m=7
T18 for (d,l,s)=(7,1,14)
42 15 9 4 6 127 −314?
26 20 16 16 214 −227?
42 17 9 8 6 36 −135T8 (ii) for 2-(7,3,2)
T11 for m=2
T18 for (d,l,s)=(2,2,6)
24 16 12 16 035 −46T10 for m=2
T12
T18 for (d,l,s)=(4,4,6)
42 18 9 6 9 035 −36T10 for m=3
T12
T18 for (d,l,s)=(3,3,6)
23 14 13 12 26 −135T11 for m=3
T18 for (d,l,s)=(3,3,6)
42 18 14 7 8 221 −320?
23 19 12 13 220 −321?
42 19 11 8 9 121 −220?
22 14 11 12 120 −221?
42 20 10 9 10 021 −120T5
T12
T18 for (d,l,s)=(1,10,2)
T20
21 11 10 11 020 −121T18 for (d,l,s)=(1,10,2)
T20
42 20 11 10 9 27 −134T11 for m=3
21 12 9 12 034 −37T10 for m=3
42 20 13 12 7 63 −138T11 for m=7
21 14 7 14 038 −73T10 for m=7
44 10 4 3 2 211 −132M9
33 27 24 27 032 −311
44 11 3 2 3 032 −111T24
32 24 23 24 011 −132
44 12 8 1 4 132 −411?
31 27 22 21 311 −232?
44 17 13 4 8 133 −510?
26 22 16 14 410 −233?
44 20 10 8 10 033 −210T10 for m=2
T12
T18 for (d,l,s)=(2,5,4)
23 13 12 12 110 −133T11 for m=2
T18 for (d,l,s)=(2,5,4)
44 20 16 10 8 411 −232?
23 19 10 14 132 −511?
44 21 11 10 10 111 −132T4
T6
T11 for m=2
22 12 10 12 032 −211T10 for m=2
T18 for (a,b,r)=(1,1,11)
45 8 4 3 1 39 −135T7 for GQ(2,2)
36 32 28 32 035 −49
45 9 2 1 2 035 −19T24
35 28 27 28 09 −135
45 12 4 1 4 040 −34does not exist by N9
32 24 23 22 24 −140
45 13 6 3 4 125 −219M17
31 24 21 22 119 −225
45 14 6 5 4 210 −134M9
M12
30 22 19 22 034 −310T18 for (a,b,r)=(2,1,5)
45 14 8 7 3 55 −139T11 for m=3
30 24 18 24 039 −65T10 for m=3
45 15 6 3 6 039 −35T10 for m=3
T25
29 20 19 18 25 −139T11 for m=3
45 16 8 5 6 124 −220M10
M12
28 20 17 18 120 −224
45 16 12 3 7 134 −510?
28 24 18 16 410 −234?
45 17 8 7 6 29 −135T11 for m=3
27 18 15 18 035 −39T10 for m=3
45 18 8 6 8 035 −29M17
26 16 15 15 19 −135
45 18 16 7 7 319 −325?
26 24 14 16 225 −419?
45 19 16 5 10 135 −69?
25 22 15 12 59 −235?
45 20 12 11 7 54 −140
24 16 10 16 040 −64does not exist by N9
46 20 15 8 9 223 −322?
25 20 13 14 222 −323?
46 21 12 9 10 123 −222?
24 15 12 13 122 −223?
46 22 11 10 11 023 −122T5
T12
T18 for (d,l,s)=(1,11,2)
23 12 11 12 022 −123T18 for (d,l,s)=(1,11,2)
T20
48 7 2 1 1 120 −127T4
40 35 33 35 027 −220
48 9 5 0 2 133 −314?
38 34 30 30 214 −233?
48 10 6 2 2 221 −226M10
37 33 28 30 126 −321
48 11 5 4 2 39 −138T9 for pg(4,3,2)
T11 for m=2
T19
M12
36 30 26 30 038 −49T10 for m=2
T18 for (a,b,r)=(3,1,3)
48 12 4 0 4 044 −43T8(i) for 2-(4,2,4)
T10 for m=2
T12
T17
T18 for (d,l,s)=(4,1,12)
35 27 26 24 33 −144T8 (ii) for 2-(4,3,8)
T11 for m=2
T18 for (d,l,s)=(8,2,12)
48 13 7 2 4 132 −315M10
34 28 24 24 215 −232
48 14 8 4 4 220 −227T19
33 27 22 24 127 −320
48 15 5 4 5 032 −115T12
T18 for (d,l,s)=(1,5,3)
T20
32 22 21 22 015 −132T18 for (d,l,s)=(2,10,3)
48 15 7 6 4 38 −139T11 for m=2
32 24 20 24 039 −48T10 for m=2
48 16 6 4 6 039 −28T10 for m=2
31 21 20 20 18 −139T11 for m=2
48 16 8 0 8 045 −82T8(i) for 2-(3,2,8)
T10 for m=2
T12
T18 for (d,l,s)=(8,1,16)
31 23 22 16 72 −145T8 (ii) for 2-(3,2,8)
T11 for m=2
T18 for (d,l,s)=(8,1,16)
48 17 7 6 6 115 −132T11 for m=2
T18 for (d,l,s)=(1,5,3)
30 20 18 20 032 −215T10 for m=2
T12
T18 for (d,l,s)=(2,10,3)
48 17 12 11 3 93 −144
30 25 15 25 044 −103does not exist by N4
48 18 9 3 9 044 −63does not exist by N4
29 20 19 15 53 −144
48 19 13 8 7 315 −232?
28 22 15 18 132 −415?
48 19 14 5 9 136 −511?
28 23 17 15 411 −236?
48 20 10 6 10 042 −45does not exist by N13
27 17 16 14 35 −142
48 20 17 7 9 228 −419?
27 24 15 15 319 −328?
48 21 13 8 10 130 −317
26 18 14 14 217 −230M8
48 21 18 9 9 320 −327?
26 23 13 15 227 −420?
48 22 11 9 11 036 −211M17
25 14 13 13 111 −136
48 22 14 10 10 218 −229M8
25 17 12 14 129 −318
48 22 17 11 9 412 −235?
25 20 11 15 135 −512?
48 23 12 11 11 112 −135T4
T6
24 13 11 13 035 −212T18 for (a,b,r)=(1,1,12)
48 23 13 12 10 36 −141T11 for m=2
24 14 10 14 041 −46T10 for m=2
M7
48 23 14 13 9 54 −143T11 for m=3
24 15 9 15 043 −64T10 for m=3
48 23 15 14 8 73 −144T8 (ii) for 2-(4,2,4)
T11 for m=2
T18 for (d,l,s)=(4,1,12)
24 16 8 16 044 −83T8(i) for 2-(4,3,8)
T10 for m=2
T12
T18 for (d,l,s)=(8,2,12)
M7
48 23 17 16 6 112 −145T11 for m=2
24 18 6 18 045 −122T10 for m=2
49 13 8 7 2 65 −143
35 30 23 30 043 −75does not exist by N13
49 15 12 1 6 139 −69does not exist by AbsBd
33 30 23 20 59 −239does not exist by AbsBd
49 19 16 9 6 511 −237?
29 26 15 20 137 −611?
49 20 12 11 6 64 −144
28 20 13 20 044 −74does not exist by N9
49 21 12 5 12 045 −73does not exist by N4
27 18 17 12 63 −145
49 22 18 7 12 138 −610?
26 22 15 12 510 −238?
50 9 5 4 1 48 −141T7 for GQ(1,4)
T19
M12
40 36 31 36 041 −58T18 for (a,b,r)=(4,1,2)
50 11 7 0 3 137 −412?
38 34 29 28 312 −237?
50 12 9 2 3 227 −322?
37 34 27 28 222 −327?
50 13 9 4 3 317 −232T16
T19
36 32 25 28 132 −417
50 14 7 6 3 47 −142?
35 28 23 28 042 −57?
50 15 6 1 6 046 −53does not exist by N4
34 25 24 21 43 −146
50 16 10 3 6 136 −413M10
33 27 22 21 313 −236
50 17 12 5 6 226 −323?
32 27 20 21 223 −326?
50 18 12 7 6 316 −233T21
31 25 18 21 133 −416
50 19 10 9 6 46 −143M17
30 21 16 21 043 −56
50 19 13 12 4 93 −146
30 24 14 24 046 −103does not exist by N4
50 20 10 5 10 045 −54T10 for m=5
T12
T18 for (d,l,s)=(5,2,10)
29 19 18 15 44 −145T11 for m=5
T18 for (d,l,s)=(5,2,10)
50 20 12 2 12 047 −102does not exist by N3
29 21 20 12 92 −147
50 21 14 7 10 135 −414?
28 21 16 15 314 −235?
50 22 16 9 10 225 −324M4
27 21 14 15 224 −325
50 23 13 10 11 125 −224M10
26 16 13 14 124 −225
50 23 16 11 10 315 −234?
26 19 12 15 134 −415?
50 24 12 11 12 025 −124T3
T5
T12
T18 for (d,l,s)=(1,12,2)
25 13 12 13 024 −125T18 for (d,l,s)=(1,12,2)
T20
50 24 14 13 10 45 −144T11 for m=5
25 15 10 15 044 −55T10 for m=5

Table 51-60

v k t λ µ rf sg comments
51 15 10 5 4 317 −233?
35 30 23 26 133 −417?
51 16 6 5 5 117 −133T4
34 24 22 24 033 −217
51 17 6 5 6 033 −117T24
33 22 21 22 017 −133
51 18 10 5 7 133 −317M17
32 24 20 20 217 −233
52 12 3 2 3 039 −112T8(i) for 2-(13,4,1)
T12
T18 for (d,l,s)=(1,3,4)
39 30 29 30 012 −139T18 for (d,l,s)=(3,9,4)
52 15 6 5 4 212 −139T8 (ii) for 2-(13,4,1)
T9 for pg(4,4,4)
T18 for (d,l,s)=(1,3,4)
36 27 24 27 039 −312T12
T18 for (d,l,s)=(3,9,4)
52 18 15 8 5 512 −239?
33 30 19 24 139 −612?
52 21 15 6 10 139 −512?
30 24 18 16 412 −239?
52 24 12 10 12 039 −212T10 for m=2
T12
T18 for (d,l,s)=(2,6,4)
27 15 14 14 112 −139T11 for m=2
T18 for (d,l,s)=(2,6,4)
52 24 18 12 10 413 −238?
27 21 12 16 138 −513?
52 25 13 12 12 113 −138T4
T6
T11 for m=2
26 14 12 14 038 −213T10 for m=2
T18 for (a,b,r)=(1,1,13)
54 7 3 0 1 133 −220does not exist by N12
46 42 39 40 120 −233
54 8 3 2 1 215 −138M10
45 40 37 40 038 −315
54 10 4 1 2 132 −221M16
43 37 34 35 121 −232
54 11 4 3 2 214 −139M16
42 35 32 35 039 −314
54 13 9 0 4 142 −511does not exist by N6
does not exist by N10
does not exist by N11
40 36 30 28 411 −242
54 14 12 2 4 233 −420?
39 37 28 28 320 −333?
54 15 5 2 5 048 −35does not exist by N13
38 28 27 26 25 −148
54 15 13 4 4 324 −329?
38 36 26 28 229 −424?
54 16 7 4 5 130 −223M16
37 28 25 26 123 −230
54 16 12 6 4 415 −238M10
37 33 24 28 138 −515
54 17 7 6 5 212 −141
36 26 23 26 041 −312T18 for (a,b,r)=(2,1,6)
54 17 9 8 4 56 −147T11 for m=2
36 28 22 28 047 −66T10 for m=2
54 17 11 10 3 84 −149T11 for m=3
36 30 21 30 049 −94T10 for m=3
54 18 7 4 7 047 −36T17
T18 for (a,b,r)=(1,2,3)
35 24 23 22 26 −147
54 18 8 2 8 050 −63does not exist by N4
35 25 24 20 53 −150
54 18 9 0 9 051 −92T8(i) for 2-(3,2,9)
T10 for m=3
T12
T18 for (d,l,s)=(9,1,18)
35 26 25 18 82 −151T8 (ii) for 2-(3,2,9)
T11 for m=3
T18 for (d,l,s)=(9,1,18)
54 19 9 6 7 129 −224M10
34 24 21 22 124 −229
54 19 13 4 8 141 −512?
34 28 22 20 412 −241?
54 20 9 8 7 211 −142M17
33 22 19 22 042 −311
54 20 16 6 8 232 −421M10
33 29 20 20 321 −332
54 21 17 8 8 323 −330M10
32 28 18 20 230 −423
54 22 16 10 8 414 −239?
31 25 16 20 139 −514?
54 23 13 12 8 55 −148
30 20 14 20 048 −65does not exist by N13
54 23 22 9 10 327 −426?
30 29 16 17 326 −427?
54 24 12 9 12 045 −38T10 for m=3
T12
T18 for (d,l,s)=(3,4,6)
29 17 16 15 28 −145T11 for m=3
T18 for (d,l,s)=(3,4,6)
54 24 16 4 16 051 −122does not exist by N3
29 21 20 10 112 −151
54 24 17 10 11 227 −326?
29 22 15 16 226 −327?
54 25 14 11 12 127 −226M10
28 17 14 15 126 −227
54 26 13 12 13 027 −126T5
T12
T18 for (d,l,s)=(1,13,2)
T20
27 14 13 14 026 −127T18 for (d,l,s)=(1,13,2)
T20
54 26 14 13 12 29 −144T11 for m=3
27 15 12 15 044 −39T10 for m=3
54 26 17 16 9 83 −150T11 for m=3
27 18 9 18 050 −93T10 for m=3
55 10 2 1 2 044 −110T12
T18 for (d,l,s)=(1,2,5)
44 36 35 36 010 −144T18 for (d,l,s)=(4,8,5)
55 14 6 5 3 310 −144T9 for pg(5,3,3)
T18 for (d,l,s)=(1,2,5)
40 32 28 32 044 −410T12
T18 for (d,l,s)=(4,8,5)
55 16 12 1 6 144 −610?
38 34 27 24 510 −244?
55 20 8 6 8 044 −210T8(i) for 2-(11,5,2)
T12
T18 for (d,l,s)=(2,4,5)
34 22 21 21 110 −144T18 for (d,l,s)=(3,6,5)
55 24 12 11 10 210 −144T8 (ii) for 2-(11,5,2)
T18 for (d,l,s)=(2,4,5)
30 18 15 18 044 −310T12
T18 for (d,l,s)=(3,6,5)
55 26 22 9 15 144 −710?
28 24 16 12 610 −244?
56 7 1 0 1 048 −17T1
T8(i) for 2-(8,2,1)
T12
T18 for (d,l,s)=(1,1,7)
48 42 41 42 07 −148T8 (ii) for 2-(8,7,6)
T18 for (d,l,s)=(6,6,7)
56 12 3 1 3 049 −26does not exist by N13
43 34 33 33 16 −149
56 13 4 3 3 121 −134T4
42 33 31 33 034 −221
56 13 7 6 2 57 −148T8 (ii) for 2-(8,2,1)
T18 for (d,l,s)=(1,1,7)
42 36 30 36 048 −67T8(i) for 2-(8,7,6)
T12
T18 for (d,l,s)=(6,6,7)
56 14 4 2 4 048 −27T10 for m=2
T12
T18 for (d,l,s)=(2,2,7)
41 31 30 30 17 −148T11 for m=2
T18 for (d,l,s)=(5,5,7)
56 15 5 4 4 120 −135T11 for m=2
40 30 28 30 035 −220T10 for m=2
56 16 13 3 5 234 −421?
39 36 27 27 321 −334?
56 19 10 9 5 56 −149
36 27 21 27 049 −66does not exist by N13
56 20 10 9 6 47 −148T18 for (d,l,s)=(2,2,7)
35 25 20 25 048 −57T12
T18 for (d,l,s)=(5,5,7)
56 21 9 6 9 048 −37T8(i) for 2-(8,4,3)
T12
T18 for (d,l,s)=(3,3,7)
34 22 21 20 27 −148T18 for (d,l,s)=(4,4,7)
56 23 16 7 11 142 −513?
32 25 19 17 413 −242?
56 24 12 8 12 049 −46T8(i) for 2-(7,4,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,3,8)
31 19 18 16 36 −149T8 (ii) for 2-(7,4,4)
T11 for m=2
T18 for (d,l,s)=(4,3,8)
56 25 15 10 12 135 −320M17
30 20 16 16 220 −235
56 26 13 11 13 042 −213M17
29 16 15 15 113 −142
56 26 16 12 12 221 −234M17
29 19 14 16 134 −321
56 26 19 13 11 414 −241M17
29 22 13 17 141 −514
56 27 14 13 13 114 −141T4
T6
28 15 13 15 041 −214T18 for (a,b,r)=(1,1,14)
56 27 15 14 12 37 −148T8 (ii) for 2-(8,4,3)
T11 for m=2
T18 for (d,l,s)=(3,3,7)
28 16 12 16 048 −47T10 for m=2
T12
T18 for (d,l,s)=(4,4,7)
56 27 20 19 7 132 −153T11 for m=7
28 21 7 21 053 −142T10 for m=7
57 16 8 3 5 138 −318?
40 32 28 28 218 −238?
57 18 6 5 6 038 −118T12
T18 for (d,l,s)=(1,6,3)
38 26 25 26 018 −138T18 for (d,l,s)=(2,12,3)
57 20 8 7 7 118 −138T18 for (d,l,s)=(1,6,3)
36 24 22 24 038 −218T12
T18 for (d,l,s)=(2,12,3)
57 22 14 9 8 318 −238M17
34 26 19 22 138 −418
58 25 23 10 11 329 −428?
32 30 17 18 328 −429?
58 26 18 11 12 229 −328M17
31 23 16 17 228 −329
58 27 15 12 13 129 −228M17
30 18 15 16 128 −229
58 28 14 13 14 029 −128T3
T5
T12
T18 for (d,l,s)=(1,14,2)
29 15 14 15 028 −129T18 for (d,l,s)=(1,14,2)
T20
60 10 2 0 2 054 −25T8(i) for 2-(6,2,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,1,10)
49 41 40 40 15 −154T8 (ii) for 2-(6,5,8)
T11 for m=2
T18 for (d,l,s)=(8,4,10)
60 11 3 2 2 124 −135T4
T11 for m=2
48 40 38 40 035 −224T10 for m=2
60 12 3 0 3 055 −34T8(i) for 2-(5,2,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,1,12)
47 38 37 36 24 −155T8 (ii) for 2-(5,4,9)
T11 for m=3
T18 for (d,l,s)=(9,3,12)
60 13 5 2 3 135 −224M10
46 38 35 36 124 −235
60 14 5 4 3 215 −144T11 for m=3
45 36 33 36 044 −315T10 for m=3
60 15 4 3 4 044 −115T20
T20
44 33 32 33 015 −144
60 15 5 0 5 056 −53T8(i) for 2-(4,2,5)
T10 for m=5
T12
T18 for (d,l,s)=(5,1,15)
44 34 33 30 43 −156T8 (ii) for 2-(4,3,10)
T11 for m=5
T18 for (d,l,s)=(10,2,15)
60 16 9 2 5 144 −415?
43 36 31 30 315 −244?
60 17 11 4 5 232 −327?
42 36 29 30 227 −332?
60 18 6 4 6 050 −29T8(i) for 2-(10,4,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,3,6)
M1 from srg(10,6,3)
41 29 28 28 19 −150T11 for m=2
T18 for (d,l,s)=(4,6,6)
60 18 11 6 5 320 −239M15
41 34 27 30 139 −420
60 19 7 6 6 120 −139T4
T11 for m=2
40 28 26 28 039 −220T10 for m=2
60 19 9 8 5 48 −151T11 for m=5
40 30 25 30 051 −58T10 for m=5
60 19 11 10 4 75 −154T8 (ii) for 2-(6,2,2)
T11 for m=2
T18 for (d,l,s)=(2,1,10)
40 32 24 32 054 −85T8(i) for 2-(6,5,8)
T10 for m=2
T12
T18 for (d,l,s)=(8,4,10)
60 20 7 6 7 039 −120T20
T20
39 26 25 26 020 −139
60 20 8 4 8 054 −45T8(i) for 2-(6,3,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,2,10)
39 27 26 24 35 −154T8 (ii) for 2-(6,4,6)
T11 for m=2
T18 for (d,l,s)=(6,3,10)
60 20 10 0 10 057 −102T8(i) for 2-(3,2,10)
T10 for m=2
T12
T18 for (d,l,s)=(10,1,20)
39 29 28 20 92 −157T8 (ii) for 2-(3,2,10)
T11 for m=2
T18 for (d,l,s)=(10,1,20)
60 21 11 6 8 139 −320M14
38 28 24 24 220 −239
60 22 12 8 8 224 −235M14
37 27 22 24 135 −324
60 22 19 6 9 239 −520?
37 34 23 22 420 −339?
60 23 11 10 8 39 −150T8 (ii) for 2-(10,4,2)
T11 for m=2
T18 for (d,l,s)=(2,3,6)
36 24 20 24 050 −49T10 for m=2
T12
T18 for (d,l,s)=(4,6,6)
60 23 14 13 6 84 −155T8 (ii) for 2-(5,2,3)
T11 for m=3
T18 for (d,l,s)=(3,1,12)
36 27 18 27 055 −94T8(i) for 2-(5,4,9)
T10 for m=3
T12
T18 for (d,l,s)=(9,3,12)
60 24 10 9 10 035 −124M14
35 21 20 21 024 −135
60 24 12 6 12 055 −64T8(i) for 2-(5,3,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,2,12)
35 23 22 18 54 −155T8 (ii) for 2-(5,3,6)
T11 for m=2
T18 for (d,l,s)=(6,2,12)
60 25 17 8 12 145 −514M14
34 26 20 18 414 −245
60 26 20 10 12 235 −424M10
33 27 18 18 324 −335
60 27 21 12 12 325 −334M14
32 26 16 18 234 −425
60 28 14 12 14 045 −214T10 for m=2
T12
T18 for (d,l,s)=(2,7,4)
31 17 16 16 114 −145T11 for m=2
T18 for (d,l,s)=(2,7,4)
60 28 20 14 12 415 −244M14
31 23 14 18 144 −515
60 29 15 14 14 115 −144T4
T6
T11 for m=2
30 16 14 16 044 −215T10 for m=2
T18 for (a,b,r)=(1,1,15)
M7
60 29 17 16 12 55 −154T8 (ii) for 2-(6,3,4)
T11 for m=2
T18 for (d,l,s)=(4,2,10)
30 18 12 18 054 −65T8(i) for 2-(6,4,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,3,10)
60 29 19 18 10 93 −156T8 (ii) for 2-(4,2,5)
T11 for m=2
T18 for (d,l,s)=(5,1,15)
30 20 10 20 056 −103T8(i) for 2-(4,3,10)
T10 for m=2
T12
T18 for (d,l,s)=(10,2,15)
M7

Table 61-70

v k t λ µ rf sg comments
62 27 24 11 12 331 −430?
34 31 18 19 330 −431?
62 28 19 12 13 231 −330?
33 24 17 18 230 −331?
62 29 16 13 14 131 −230?
32 19 16 17 130 −231?
62 30 15 14 15 031 −130T5
T12
T18 for (d,l,s)=(1,15,2)
31 16 15 16 030 −131T18 for (d,l,s)=(1,15,2)
T20
63 8 2 1 1 127 −135T2
T4
54 48 46 48 035 −227
63 11 8 1 2 235 −327T22
51 48 41 42 227 −335
63 16 8 4 4 227 −235?
46 38 33 35 135 −327?
63 16 10 1 5 149 −513?
46 40 34 32 413 −249?
63 18 6 3 6 056 −36T8(i) for 2-(7,3,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,2,9)
44 32 31 30 26 −156T11 for m=3
T18 for (d,l,s)=(6,4,9)
63 18 14 5 5 328 −334?
44 40 30 32 234 −428?
63 19 8 5 6 135 −227M17
43 32 29 30 127 −235
63 20 8 7 6 214 −148T11 for m=3
42 30 27 30 048 −314T10 for m=3
T18 for (a,b,r)=(2,1,7)
63 20 10 9 5 57 −155M17
42 32 26 32 055 −67
63 21 8 5 8 055 −37T25
41 28 27 26 27 −155
63 22 10 7 8 134 −228M10
40 28 25 26 128 −234
63 22 14 5 9 148 −514?
40 32 26 24 414 −248?
63 23 10 9 8 213 −149M17
39 26 23 26 049 −313
63 24 18 9 9 327 −335?
38 32 22 24 235 −427?
63 26 14 13 9 56 −156T8 (ii) for 2-(7,3,3)
T11 for m=3
T18 for (d,l,s)=(3,2,9)
36 24 18 24 056 −66T10 for m=3
T12
T18 for (d,l,s)=(6,4,9)
63 27 12 11 12 035 −127M17
35 20 19 20 027 −135
64 13 6 1 3 144 −319?
50 43 39 39 219 −244?
64 14 7 3 3 228 −235M17
49 42 37 39 135 −328
64 15 6 5 3 312 −151T9 for pg(4,4,3)
T19
M12
48 39 35 39 051 −412T18 for (a,b,r)=(3,1,4)
64 15 9 8 2 76 −157T11 for m=2
48 42 34 42 057 −86T10 for m=2
64 16 5 1 5 059 −44does not exist by N9
47 36 35 33 34 −159
64 17 8 3 5 143 −320M17
46 37 33 33 220 −243
64 17 13 0 6 153 −710does not exist by N5
does not exist by AbsBd
does not exist by N10
does not exist by N11
46 42 34 30 610 −253does not exist by AbsBd
64 18 9 5 5 227 −236T19
45 36 31 33 136 −327
64 19 8 7 5 311 −152M17
44 33 29 33 052 −411
64 22 18 10 6 613 −250M17
41 37 24 30 150 −713
64 23 13 12 6 75 −158
40 30 22 30 058 −85does not exist by N13
64 24 12 4 12 060 −83does not exist by N4
39 27 26 20 73 −160
64 25 19 6 12 152 −711?
38 32 24 20 611 −252?
64 26 24 8 12 244 −619?
37 35 22 20 519 −344?
64 27 18 9 13 148 −515?
36 27 21 19 415 −248?
64 27 18 17 7 113 −160
36 27 15 27 060 −123does not exist by N4
64 28 14 10 14 056 −47T10 for m=2
M17
35 21 20 18 37 −156T11 for m=2
64 29 17 12 14 140 −323
34 22 18 18 223 −240M8
64 29 27 14 12 520 −343?
34 32 16 20 243 −620?
64 30 15 13 15 048 −215M17
33 18 17 17 115 −148
64 30 18 14 14 224 −239M8
33 21 16 18 139 −324
64 30 21 15 13 416 −247M17
33 24 15 19 147 −516
64 30 24 16 12 612 −251?
33 27 14 20 151 −712?
64 31 16 15 15 116 −147T4
T6
32 17 15 17 047 −216T18 for (a,b,r)=(1,1,16)
64 31 17 16 14 38 −155T11 for m=2
32 18 14 18 055 −48T10 for m=2
64 31 19 18 12 74 −159T11 for m=2
32 20 12 20 059 −84T10 for m=2
M7
64 31 23 22 8 152 −161T11 for m=2
32 24 8 24 061 −162T10 for m=2
M7
65 11 4 1 2 139 −225?
53 46 43 44 125 −239?
65 14 4 3 3 125 −139?
50 40 38 40 039 −225?
65 22 16 6 8 239 −425?
42 36 27 27 325 −339?
65 25 10 9 10 039 −125M17
39 24 23 24 025 −139
65 28 16 12 12 225 −239M17
36 24 19 21 139 −325
66 10 5 4 1 411 −154?
55 50 45 50 054 −511?
66 11 2 1 2 054 −111T24
54 45 44 45 011 −154
66 17 15 2 5 244 −521?
48 46 35 34 421 −344?
66 19 9 4 6 144 −321?
46 36 32 32 221 −244?
66 21 7 6 7 044 −121T12
T18 for (d,l,s)=(1,7,3)
44 30 29 30 021 −144T18 for (d,l,s)=(2,14,3)
66 21 9 8 6 311 −154T11 for m=2
44 32 28 32 054 −411T10 for m=2
66 22 8 6 8 054 −211T10 for m=2
43 29 28 28 111 −154T11 for m=2
66 22 11 0 11 063 −112T8(i) for 2-(3,2,11)
T10 for m=11
T12
T18 for (d,l,s)=(11,1,22)
43 32 31 22 102 −163T8 (ii) for 2-(3,2,11)
T11 for m=11
T18 for (d,l,s)=(11,1,22)
66 23 9 8 8 121 −144T11 for m=2
T18 for (d,l,s)=(1,7,3)
42 28 26 28 044 −221T10 for m=2
T12
T18 for (d,l,s)=(2,14,3)
66 23 17 4 10 154 −711?
42 36 28 24 611 −254?
66 25 15 10 9 321 −244?
40 30 23 26 144 −421?
66 25 20 5 12 155 −810does not exist by AbsBd
40 35 26 21 710 −255does not exist by AbsBd
66 27 25 12 10 521 −344?
38 36 20 24 244 −621?
66 29 25 12 13 333 −432?
36 32 19 20 332 −433?
66 30 15 12 15 055 −310T8(i) for 2-(11,6,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,5,6)
35 20 19 18 210 −155T8 (ii) for 2-(11,6,3)
T11 for m=3
T18 for (d,l,s)=(3,5,6)
66 30 20 13 14 233 −332?
35 25 18 19 232 −333?
66 31 17 14 15 133 −232M17
34 20 17 18 132 −233
66 31 26 17 12 711 −254?
34 29 14 21 154 −811?
66 32 16 15 16 033 −132T5
T12
T18 for (d,l,s)=(1,16,2)
T20
33 17 16 17 032 −133T18 for (d,l,s)=(1,16,2)
T20
66 32 17 16 15 211 −154T11 for m=3
33 18 15 18 054 −311T10 for m=3
66 32 21 20 11 103 −162T11 for m=11
33 22 11 22 062 −113T10 for m=11
68 13 7 0 3 151 −416?
54 48 43 42 316 −251?
68 16 4 3 4 051 −116T12
T18 for (d,l,s)=(1,4,4)
51 39 38 39 016 −151T18 for (d,l,s)=(3,12,4)
68 19 7 6 5 216 −151T18 for (d,l,s)=(1,4,4)
48 36 33 36 051 −316T12
T18 for (d,l,s)=(3,12,4)
68 22 16 9 6 516 −251M17
45 39 28 33 151 −616
68 29 19 10 14 151 −516?
38 28 22 20 416 −251?
68 32 16 14 16 051 −216T10 for m=2
T12
T18 for (d,l,s)=(2,8,4)
35 19 18 18 116 −151T11 for m=2
T18 for (d,l,s)=(2,8,4)
68 32 22 16 14 417 −250?
35 25 16 20 150 −517?
68 33 17 16 16 117 −150T4
T6
T11 for m=2
34 18 16 18 050 −217T10 for m=2
T18 for (a,b,r)=(1,1,17)
69 21 12 7 6 323 −245?
47 38 31 34 145 −423?
69 22 8 7 7 123 −145T4
46 32 30 32 045 −223
69 23 8 7 8 045 −123M17
45 30 29 30 023 −145
69 24 12 7 9 145 −323?
44 32 28 28 223 −245?
69 25 20 7 10 245 −523?
43 38 27 26 423 −345?
70 9 3 2 1 220 −149?
60 54 51 54 049 −320?
70 11 5 0 2 149 −320?
58 52 48 48 220 −249?
70 13 5 4 2 314 −155T11 for m=2
56 48 44 48 055 −414T10 for m=2
70 14 3 2 3 055 −114M17
55 44 43 44 014 −155
70 15 9 0 4 155 −514does not exist by N10
does not exist by N11
54 48 42 40 414 −255
70 18 12 6 4 420 −249?
51 45 36 40 149 −520?
70 20 6 5 6 049 −120M17
49 35 34 35 020 −149
70 22 20 4 8 249 −620?
47 45 32 30 520 −349?
70 26 20 12 8 614 −255?
43 37 24 30 155 −714?
70 27 12 11 10 214 −155M17
42 27 24 27 055 −314
70 28 12 10 12 055 −214T10 for m=2
41 25 24 24 114 −155T11 for m=2
70 28 14 7 14 065 −74T10 for m=7
T12
T18 for (d,l,s)=(7,2,14)
41 27 26 21 64 −165T11 for m=7
T18 for (d,l,s)=(7,2,14)
70 29 13 12 12 120 −149T11 for m=2
40 24 22 24 049 −220T10 for m=2
70 29 20 9 14 155 −614?
40 31 24 21 514 −255?
70 30 15 10 15 063 −56T10 for m=5
T12
T18 for (d,l,s)=(5,3,10)
39 24 23 20 46 −163T11 for m=5
T18 for (d,l,s)=(5,3,10)
70 30 24 11 14 245 −524?
39 33 22 21 424 −345?
70 31 19 12 15 149 −420?
38 26 21 20 320 −249?
70 31 26 13 14 335 −434?
38 33 20 21 334 −435?
70 32 21 14 15 235 −334?
37 26 19 20 234 −335?
70 32 26 15 14 425 −344?
37 31 18 21 244 −525?
70 33 18 15 16 135 −234T20
36 21 18 19 134 −235
70 33 21 16 15 321 −248?
36 24 17 20 148 −421?
70 33 24 17 14 515 −254?
36 27 16 21 154 −615?
70 34 17 16 17 035 −134T5
T12
T18 for (d,l,s)=(1,17,2)
35 18 17 18 034 −135T18 for (d,l,s)=(1,17,2)
T20
70 34 19 18 15 47 −162T11 for m=5
35 20 15 20 062 −57T10 for m=5
70 34 20 19 14 65 −164T11 for m=7
35 21 14 21 064 −75T10 for m=7

Table 71-80

v k t λ µ rf sg comments
72 8 1 0 1 063 −18T1
T8(i) for 2-(9,2,1)
T12
T18 for (d,l,s)=(1,1,8)
63 56 55 56 08 −163T8 (ii) for 2-(9,8,7)
T18 for (d,l,s)=(7,7,8)
72 11 6 5 1 510 −161T7 for GQ(1,5)
60 55 49 55 061 −610T18 for (a,b,r)=(5,1,2)
72 14 11 1 3 245 −426?
57 54 45 45 326 −345?
72 15 8 7 2 68 −163T8 (ii) for 2-(9,2,1)
T18 for (d,l,s)=(1,1,8)
56 49 42 49 063 −78T8(i) for 2-(9,8,7)
T12
T18 for (d,l,s)=(7,7,8)
72 15 12 3 3 333 −338?
56 53 43 45 238 −433?
72 16 4 2 4 063 −28T8(i) for 2-(9,3,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,2,8)
55 43 42 42 18 −163T11 for m=2
T18 for (d,l,s)=(6,6,8)
72 16 11 5 3 421 −250T26
55 50 41 45 150 −521
72 17 5 4 4 127 −144T4
T11 for m=2
54 42 40 42 044 −227T10 for m=2
72 17 8 7 3 59 −162T11 for m=3
54 45 39 45 062 −69T10 for m=3
72 18 5 3 5 062 −29M17
53 40 39 39 19 −162
72 18 6 0 6 068 −63T8(i) for 2-(4,2,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,1,18)
53 41 40 36 53 −168T8 (ii) for 2-(4,3,12)
T11 for m=2
T18 for (d,l,s)=(12,2,18)
72 19 6 5 5 126 −145T20
52 39 37 39 045 −226
72 19 11 2 6 156 −515M10
52 44 38 36 415 −256
72 20 14 4 6 244 −427M10
51 45 36 36 327 −344
72 21 7 4 7 064 −37?
50 36 35 34 27 −164?
72 21 15 6 6 332 −339M10
50 44 34 36 239 −432
72 22 9 6 7 140 −231M10
49 36 33 34 131 −240
72 22 14 8 6 420 −251T26
49 41 32 36 151 −520
72 23 9 8 7 216 −155
48 34 31 34 055 −316T18 for (a,b,r)=(2,1,8)
72 23 11 10 6 58 −163T8 (ii) for 2-(9,3,2)
T11 for m=2
T18 for (d,l,s)=(2,2,8)
48 36 30 36 063 −68T10 for m=2
T12
T18 for (d,l,s)=(6,6,8)
72 23 15 14 4 114 −167T11 for m=2
48 40 28 40 067 −124T10 for m=2
72 24 9 6 9 063 −38T8(i) for 2-(9,4,3)
T10 for m=3
T12
T18 for (a,b,r)=(1,2,4)
T18 for (d,l,s)=(3,3,8)
47 32 31 30 28 −163T8 (ii) for 2-(9,6,5)
T11 for m=3
T18 for (d,l,s)=(5,5,8)
72 24 10 4 10 067 −64T10 for m=2
47 33 32 28 54 −167T11 for m=2
72 24 12 0 12 069 −122T8(i) for 2-(3,2,12)
T10 for m=2
T12
T18 for (d,l,s)=(12,1,24)
47 35 34 24 112 −169T8 (ii) for 2-(3,2,12)
T11 for m=2
T18 for (d,l,s)=(12,1,24)
72 25 11 8 9 139 −232T20
46 32 29 30 132 −239
72 25 15 6 10 155 −516M17
46 36 30 28 416 −255
72 26 11 10 9 215 −156T11 for m=3
45 30 27 30 056 −315T10 for m=3
72 26 18 8 10 243 −428M10
45 37 28 28 328 −343
72 27 19 10 10 331 −340M17
44 36 26 28 240 −431
72 28 18 12 10 419 −252M15
43 33 24 28 152 −519
72 29 15 14 10 57 −164?
42 28 22 28 064 −67?
72 30 15 9 15 066 −65does not exist by N13
41 26 25 21 55 −166
72 31 16 15 12 48 −163T8 (ii) for 2-(9,4,3)
T18 for (d,l,s)=(3,3,8)
40 25 20 25 063 −58T8(i) for 2-(9,6,5)
T12
T18 for (d,l,s)=(5,5,8)
72 31 20 11 15 154 −517?
40 29 23 21 417 −254?
72 32 16 12 16 063 −48T10 for m=2
T12
T18 for (d,l,s)=(4,4,8)
39 23 22 20 38 −163T11 for m=2
T18 for (d,l,s)=(4,4,8)
72 32 23 13 15 242 −429M17
39 30 21 21 329 −342
72 33 19 14 16 145 −326T20
38 24 20 20 226 −245
72 33 24 15 15 330 −341M17
38 29 19 21 241 −430
72 34 17 15 17 054 −217M17
37 20 19 19 117 −154
72 34 20 16 16 227 −244T20
37 23 18 20 144 −327
72 34 23 17 15 418 −253M17
37 26 17 21 153 −518
72 35 18 17 17 118 −153T4
T6
36 19 17 19 053 −218T18 for (a,b,r)=(1,1,18)
72 35 19 18 16 39 −162T11 for m=2
36 20 16 20 062 −49T10 for m=2
M7
72 35 20 19 15 56 −165T11 for m=3
36 21 15 21 065 −66T10 for m=3
72 35 23 22 12 113 −168T8 (ii) for 2-(4,2,6)
T11 for m=2
T18 for (d,l,s)=(6,1,18)
36 24 12 24 068 −123T8(i) for 2-(4,3,12)
T10 for m=2
T12
T18 for (d,l,s)=(12,2,18)
M7
72 35 26 25 9 172 −169T11 for m=3
36 27 9 27 069 −182T10 for m=3
74 33 27 14 15 337 −436?
40 34 21 22 336 −437?
74 34 22 15 16 237 −336?
39 27 20 21 236 −337?
74 35 19 16 17 137 −236M17
38 22 19 20 136 −237
74 36 18 17 18 037 −136T3
T5
T12
T18 for (d,l,s)=(1,18,2)
37 19 18 19 036 −137T18 for (d,l,s)=(1,18,2)
T20
75 12 8 1 2 242 −332?
62 58 51 52 232 −342?
75 13 8 3 2 327 −247?
61 56 49 52 147 −427?
75 14 6 5 2 412 −162T19
M12
60 52 47 52 062 −512T18 for (a,b,r)=(4,1,3)
75 16 8 1 4 156 −418?
58 50 45 44 318 −256?
75 17 10 3 4 241 −333?
57 50 43 44 233 −341?
75 18 10 5 4 326 −248T19
56 48 41 44 148 −426
75 19 8 7 4 411 −163?
55 44 39 44 063 −511?
75 20 16 3 6 250 −524?
54 50 39 38 424 −350?
75 22 10 5 7 150 −324?
52 40 36 36 224 −250?
75 24 8 7 8 050 −124T12
T18 for (d,l,s)=(1,8,3)
50 34 33 34 024 −150T18 for (d,l,s)=(2,16,3)
75 24 14 13 5 95 −169T11 for m=5
50 40 30 40 069 −105T10 for m=5
75 25 10 5 10 069 −55T10 for m=5
49 34 33 30 45 −169T11 for m=5
75 26 10 9 9 124 −150T18 for (d,l,s)=(1,8,3)
48 32 30 32 050 −224T12
T18 for (d,l,s)=(2,16,3)
75 26 14 7 10 154 −420?
48 36 31 30 320 −254?
75 27 16 9 10 239 −335?
47 36 29 30 235 −339?
75 28 16 11 10 324 −250T21
46 34 27 30 150 −424
75 28 24 14 8 812 −262?
46 42 25 33 162 −912?
75 29 14 13 10 49 −165T11 for m=5
45 30 25 30 065 −59T10 for m=5
75 29 20 19 6 143 −171
45 36 21 36 071 −153does not exist by N4
75 30 14 9 14 068 −56M17
44 28 27 24 46 −168
75 30 16 6 16 071 −103does not exist by N4
44 30 29 21 93 −171
75 30 18 3 18 072 −152does not exist by N3
44 32 31 18 142 −172
75 30 26 13 11 524 −350?
44 40 24 28 250 −624?
75 31 18 11 14 153 −421?
43 30 25 24 321 −253?
75 32 20 13 14 238 −336T21
42 30 23 24 236 −338
75 33 20 15 14 323 −251?
41 28 21 24 151 −423?
75 34 18 17 14 48 −166M17
40 24 19 24 066 −58
75 36 30 13 21 163 −911does not exist by AbsBd
38 32 22 16 811 −263does not exist by AbsBd
76 17 13 6 3 519 −256?
58 54 43 48 156 −619?
76 18 6 5 4 219 −156M17
57 45 42 45 056 −319
76 19 5 4 5 056 −119M17
56 42 41 42 019 −156
76 20 10 3 6 156 −419?
55 45 40 39 319 −256?
76 33 21 12 16 157 −518?
42 30 24 22 418 −257?
76 36 18 16 18 057 −218T10 for m=2
T12
T18 for (d,l,s)=(2,9,4)
39 21 20 20 118 −157T11 for m=2
T18 for (d,l,s)=(2,9,4)
76 36 24 18 16 419 −256M17
39 27 18 22 156 −519
76 37 19 18 18 119 −156T4
T6
T11 for m=2
38 20 18 20 056 −219T10 for m=2
T18 for (a,b,r)=(1,1,19)
77 13 4 3 2 221 −155?
63 54 51 54 055 −321?
77 21 6 5 6 055 −121M17
55 40 39 40 021 −155
77 26 16 10 8 421 −255?
50 40 31 35 155 −521?
77 29 16 9 12 155 −421?
47 34 29 28 321 −255?
77 34 16 15 15 121 −155M17
42 24 22 24 055 −221
77 37 34 15 20 255 −721?
39 36 21 18 621 −355?
78 12 2 1 2 065 −112T8(i) for 2-(13,3,1)
T12
T18 for (d,l,s)=(1,2,6)
65 55 54 55 012 −165T18 for (d,l,s)=(5,10,6)
78 17 7 6 3 412 −165T8 (ii) for 2-(13,3,1)
T18 for (d,l,s)=(1,2,6)
60 50 45 50 065 −512T12
T18 for (d,l,s)=(5,10,6)
78 19 13 0 6 165 −712does not exist by N6
does not exist by N10
does not exist by N11
58 52 44 40 612 −265
78 23 21 8 6 526 −351?
54 52 36 40 251 −626?
78 24 8 6 8 065 −212T10 for m=2
T12
T18 for (d,l,s)=(2,4,6)
53 37 36 36 112 −165T11 for m=2
T18 for (d,l,s)=(4,8,6)
78 24 13 8 7 326 −251M17
53 42 35 38 151 −426
78 25 9 8 8 126 −151T4
T11 for m=2
52 36 34 36 051 −226T10 for m=2
78 26 9 8 9 051 −126T20
T20
51 34 33 34 026 −151
78 26 13 0 13 075 −132T8(i) for 2-(3,2,13)
T10 for m=13
T12
T18 for (d,l,s)=(13,1,26)
51 38 37 26 122 −175T8 (ii) for 2-(3,2,13)
T11 for m=13
T18 for (d,l,s)=(13,1,26)
78 27 13 8 10 151 −326?
50 36 32 32 226 −251?
78 28 21 8 11 251 −526?
49 42 31 30 426 −351?
78 29 13 12 10 312 −165T11 for m=2
T18 for (d,l,s)=(2,4,6)
48 32 28 32 065 −412T10 for m=2
T12
T18 for (d,l,s)=(4,8,6)
78 31 23 8 15 165 −812?
46 38 29 24 712 −265?
78 34 28 18 12 812 −265?
43 37 20 28 165 −912?
78 35 28 15 16 339 −438?
42 35 22 23 338 −439?
78 36 18 15 18 065 −312T10 for m=3
T12
T18 for (d,l,s)=(3,6,6)
M1 from srg(13,6,2)
41 23 22 21 212 −165T11 for m=3
T18 for (d,l,s)=(3,6,6)
78 36 23 16 17 239 −338T20
41 28 21 22 238 −339
78 37 20 17 18 139 −238T20
40 23 20 21 138 −239
78 37 29 20 15 713 −264?
40 32 17 24 164 −813?
78 38 19 18 19 039 −138T5
T12
T18 for (d,l,s)=(1,19,2)
T20
39 20 19 20 038 −139T18 for (d,l,s)=(1,19,2)
T20
78 38 20 19 18 213 −164T11 for m=3
39 21 18 21 064 −313T10 for m=3
78 38 25 24 13 123 −174T11 for m=13
39 26 13 26 074 −133T10 for m=13
80 9 2 1 1 135 −144T4
70 63 61 63 044 −235
80 14 10 4 2 424 −255?
65 61 52 56 155 −524?
80 15 3 2 3 064 −115T8(i) for 2-(16,4,1)
T12
T18 for (d,l,s)=(1,3,5)
64 52 51 52 015 −164T18 for (d,l,s)=(4,12,5)
80 16 4 0 4 075 −44T8(i) for 2-(5,2,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,1,16)
63 51 50 48 34 −175T8 (ii) for 2-(5,4,12)
T11 for m=2
T18 for (d,l,s)=(12,3,16)
80 17 7 2 4 155 −324?
62 52 48 48 224 −255?
80 18 8 4 4 235 −244M17
61 51 46 48 144 −335
80 19 7 6 4 315 −164T8 (ii) for 2-(16,4,1)
T9 for pg(5,4,4)
T11 for m=2
T18 for (d,l,s)=(1,3,5)
T19
60 48 44 48 064 −415T10 for m=2
T12
T18 for (a,b,r)=(3,1,5)
T18 for (d,l,s)=(4,12,5)
80 19 13 12 2 115 −174
60 54 42 54 074 −125does not exist by N13
80 20 6 2 6 074 −45T10 for m=2
59 45 44 42 35 −174T11 for m=2
80 21 9 4 6 154 −325M17
58 46 42 42 225 −254
80 22 10 6 6 234 −245T20
M13
57 45 40 42 145 −334
80 23 9 8 6 314 −165T11 for m=2
T20
56 42 38 42 065 −414T10 for m=2
80 23 16 15 3 134 −175
56 49 35 49 075 −144does not exist by N9
80 23 19 10 5 715 −264?
56 52 37 44 164 −815?
80 24 9 3 9 075 −64does not exist by N9
55 40 39 35 54 −175
80 26 16 5 10 164 −615?
53 43 36 33 515 −264?
80 27 18 9 9 335 −344?
52 43 33 35 244 −435?
80 30 12 10 12 064 −215T10 for m=2
T12
T18 for (d,l,s)=(2,6,5)
49 31 30 30 115 −164T11 for m=2
T18 for (d,l,s)=(3,9,5)
80 30 15 5 15 076 −103does not exist by N4
49 34 33 25 93 −176
80 31 13 12 12 124 −155T11 for m=2
48 30 28 30 055 −224T10 for m=2
80 31 19 18 8 114 −175T8 (ii) for 2-(5,2,4)
T11 for m=2
T18 for (d,l,s)=(4,1,16)
48 36 24 36 075 −124T8(i) for 2-(5,4,12)
T10 for m=2
T12
T18 for (d,l,s)=(12,3,16)
80 31 24 7 15 168 −911does not exist by AbsBd
48 41 31 25 811 −268does not exist by AbsBd
80 32 16 8 16 075 −84T10 for m=2
T12
T18 for (d,l,s)=(8,2,16)
47 31 30 24 74 −175T11 for m=2
T18 for (d,l,s)=(8,2,16)
80 33 23 10 16 165 −714?
46 36 28 24 614 −265?
80 34 16 15 14 215 −164T18 for (d,l,s)=(2,6,5)
45 27 24 27 064 −315T12
T18 for (d,l,s)=(3,9,5)
80 34 28 12 16 255 −624?
45 39 26 24 524 −355?
80 35 22 13 17 160 −519?
44 31 25 23 419 −260?
80 35 31 14 16 345 −534M17
44 40 24 24 434 −445
80 36 18 14 18 070 −49T10 for m=2
M17
43 25 24 22 39 −170T11 for m=2
80 36 32 16 16 435 −444M17
43 39 22 24 344 −535
80 37 21 16 18 150 −329
42 26 22 22 229 −250M8
80 37 31 18 16 525 −354?
42 36 20 24 254 −625?
80 38 19 17 19 060 −219M17
41 22 21 21 119 −160
80 38 22 18 18 230 −249M8
41 25 20 22 149 −330
80 38 25 19 17 420 −259M17
41 28 19 23 159 −520
80 38 28 20 16 615 −264?
41 31 18 24 164 −715?
80 38 31 21 15 812 −267does not exist by AbsBd
41 34 17 25 167 −912does not exist by AbsBd
80 39 20 19 19 120 −159T4
T6
40 21 19 21 059 −220T18 for (a,b,r)=(1,1,20)
80 39 21 20 18 310 −169T11 for m=2
40 22 18 22 069 −410T10 for m=2
M7
80 39 23 22 16 75 −174T11 for m=2
40 24 16 24 074 −85T10 for m=2
80 39 24 23 15 94 −175T11 for m=5
40 25 15 25 075 −104T10 for m=5
80 39 29 28 10 192 −177T11 for m=2
40 30 10 30 077 −202T10 for m=2
M7

Table 81-90

v k t λ µ rf sg comments
81 17 10 9 2 87 −173?
63 56 47 56 073 −97?
81 22 12 3 7 163 −517?
58 48 42 40 417 −263?
81 24 8 5 8 072 −38?
56 40 39 38 28 −172?
81 24 16 7 7 336 −344?
56 48 38 40 244 −436?
81 25 10 7 8 145 −235M17
55 40 37 38 135 −245
81 25 20 11 6 715 −265?
55 50 35 42 165 −815?
81 26 10 9 8 218 −162
54 38 35 38 062 −318T18 for (a,b,r)=(2,1,9)
81 26 12 11 7 59 −171M17
54 40 34 40 071 −69
81 26 14 13 6 86 −174T11 for m=3
54 42 33 42 074 −96T10 for m=3
81 27 10 7 10 071 −39T25
53 36 35 34 29 −171
81 27 12 3 12 077 −93does not exist by N4
53 38 37 30 83 −177
81 28 12 9 10 144 −236M13
52 36 33 34 136 −244
81 28 16 7 11 162 −518?
52 40 34 32 418 −262?
81 28 20 5 12 168 −812does not exist by AbsBd
52 44 35 30 712 −268does not exist by AbsBd
81 29 12 11 10 217 −163M17
51 34 31 34 063 −317
81 29 26 7 12 259 −721?
51 48 33 30 621 −359?
81 30 20 11 11 335 −345?
50 40 30 32 245 −435?
81 32 16 15 11 58 −172?
48 32 26 32 072 −68?
81 33 30 15 12 623 −357?
47 44 25 30 257 −723?
81 34 26 17 12 714 −266?
46 38 23 30 166 −814?
81 35 20 19 12 85 −175
45 30 21 30 075 −95does not exist by N13
81 36 20 11 20 076 −94does not exist by N9
44 28 27 20 84 −176
81 36 24 6 24 078 −182does not exist by N3
44 32 31 15 172 −178
81 37 28 13 20 167 −813?
43 34 25 20 713 −267?
81 38 34 15 20 258 −722?
42 38 23 20 622 −358?
82 37 29 16 17 341 −440M17
44 36 23 24 340 −441
82 38 24 17 18 241 −340?
43 29 22 23 240 −341?
82 39 21 18 19 141 −240M17
42 24 21 22 140 −241
82 40 20 19 20 041 −140T3
T5
T12
T18 for (d,l,s)=(1,20,2)
41 21 20 21 040 −141T18 for (d,l,s)=(1,20,2)
T20
84 9 7 0 1 248 −335does not exist by N12
74 72 65 66 235 −348
84 12 2 0 2 077 −26T8(i) for 2-(7,2,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,1,12)
71 61 60 60 16 −177T8 (ii) for 2-(7,6,10)
T11 for m=2
T18 for (d,l,s)=(10,5,12)
84 13 3 2 2 135 −148T4
T11 for m=2
70 60 58 60 048 −235T10 for m=2
84 17 8 1 4 163 −420?
66 57 52 51 320 −263?
84 20 5 4 5 063 −120T12
T18 for (d,l,s)=(1,5,4)
63 48 47 48 020 −163T18 for (d,l,s)=(3,15,4)
84 20 11 10 3 87 −176?
63 54 45 54 076 −97?
84 21 6 3 6 076 −37T10 for m=3
62 47 46 45 27 −176T11 for m=3
84 21 7 0 7 080 −73T8(i) for 2-(4,2,7)
T10 for m=7
T12
T18 for (d,l,s)=(7,1,21)
62 48 47 42 63 −180T8 (ii) for 2-(4,3,14)
T11 for m=7
T18 for (d,l,s)=(14,2,21)
84 22 8 5 6 148 −235?
61 47 44 45 135 −248?
84 22 13 2 7 168 −615?
61 52 45 42 515 −268?
84 23 8 7 6 220 −163T11 for m=3
T18 for (d,l,s)=(1,5,4)
60 45 42 45 063 −320T10 for m=3
T12
T18 for (d,l,s)=(3,15,4)
84 23 13 12 4 96 −177T8 (ii) for 2-(7,2,2)
T11 for m=2
T18 for (d,l,s)=(2,1,12)
60 50 40 50 077 −106T8(i) for 2-(7,6,10)
T10 for m=2
T12
T18 for (d,l,s)=(10,5,12)
84 23 17 4 7 256 −527?
60 54 43 42 427 −356?
84 24 8 4 8 077 −46T8(i) for 2-(7,3,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,2,12)
59 43 42 40 36 −177T11 for m=2
T18 for (d,l,s)=(8,4,12)
84 24 19 6 7 344 −439?
59 54 41 42 339 −444?
84 25 11 6 8 156 −327?
58 44 40 40 227 −256?
84 25 19 8 7 432 −351?
58 52 39 42 251 −532?
84 26 12 8 8 235 −248?
57 43 38 40 148 −335?
84 26 17 10 7 520 −263?
57 48 37 42 163 −620?
84 26 22 12 6 814 −269?
57 53 36 44 169 −914?
84 27 9 8 9 056 −127T12
T18 for (d,l,s)=(1,9,3)
T20
56 38 37 38 027 −156T18 for (d,l,s)=(2,18,3)
84 27 11 10 8 314 −169T11 for m=2
56 40 36 40 069 −414T10 for m=2
84 27 13 12 7 68 −175T11 for m=7
56 42 35 42 075 −78T10 for m=7
84 28 10 8 10 069 −214T10 for m=2
55 37 36 36 114 −169T11 for m=2
84 28 14 0 14 081 −142T8(i) for 2-(3,2,14)
T10 for m=2
T12
T18 for (d,l,s)=(14,1,28)
55 41 40 28 132 −181T8 (ii) for 2-(3,2,14)
T11 for m=2
T18 for (d,l,s)=(14,1,28)
84 29 11 10 10 127 −156T11 for m=2
T18 for (d,l,s)=(1,9,3)
54 36 34 36 056 −227T10 for m=2
T12
T18 for (d,l,s)=(2,18,3)
84 29 19 6 12 169 −714M10
54 44 36 32 614 −269
84 31 17 12 11 327 −256M10
52 38 31 34 156 −427
84 31 27 10 12 348 −535?
52 48 32 32 435 −448?
84 33 27 14 12 527 −356?
50 44 28 32 256 −627?
84 35 15 14 15 048 −135M17
48 28 27 28 035 −148
84 35 19 18 12 76 −177T8 (ii) for 2-(7,3,4)
T11 for m=2
T18 for (d,l,s)=(4,2,12)
48 32 24 32 077 −86T10 for m=2
T12
T18 for (d,l,s)=(8,4,12)
84 36 18 12 18 077 −66T8(i) for 2-(7,4,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,3,12)
47 29 28 24 56 −177T8 (ii) for 2-(7,4,6)
T11 for m=2
T18 for (d,l,s)=(6,3,12)
84 37 23 14 18 163 −520M17
46 32 26 24 420 −263
84 38 26 16 18 249 −434?
45 33 24 24 334 −349?
84 39 27 18 18 335 −348M10
44 32 22 24 248 −435
84 40 20 18 20 063 −220T10 for m=2
T12
T18 for (d,l,s)=(2,10,4)
43 23 22 22 120 −163T11 for m=2
T18 for (d,l,s)=(2,10,4)
84 40 26 20 18 421 −262M17
43 29 20 24 162 −521
84 41 21 20 20 121 −162T4
T6
T11 for m=2
42 22 20 22 062 −221T10 for m=2
T18 for (a,b,r)=(1,1,21)
M7
84 41 23 22 18 57 −176T11 for m=2
42 24 18 24 076 −67T10 for m=2
84 41 27 26 14 133 −180T8 (ii) for 2-(4,2,7)
T11 for m=2
T18 for (d,l,s)=(7,1,21)
42 28 14 28 080 −143T8(i) for 2-(4,3,14)
T10 for m=2
T12
T18 for (d,l,s)=(14,2,21)
M7
85 16 4 3 3 134 −150T4
68 56 54 56 050 −234
85 18 6 3 4 150 −234?
66 54 51 52 134 −250?
85 32 16 12 12 234 −250?
52 36 31 33 150 −334?
85 34 14 13 14 050 −134M17
50 30 29 30 034 −150
85 36 24 14 16 250 −434?
48 36 27 27 334 −350?
86 38 37 16 17 443 −542does not exist by N14
47 46 25 26 442 −543
86 39 30 17 18 343 −442?
46 37 24 25 342 −443?
86 40 25 18 19 243 −342?
45 30 23 24 242 −343?
86 41 22 19 20 143 −242?
44 25 22 23 142 −243?
86 42 21 20 21 043 −142T5
T12
T18 for (d,l,s)=(1,21,2)
43 22 21 22 042 −143T18 for (d,l,s)=(1,21,2)
T20
87 26 22 9 7 529 −357?
60 56 40 44 257 −629?
87 27 14 9 8 329 −257?
59 46 39 42 157 −429?
87 28 10 9 9 129 −157T4
58 40 38 40 057 −229
87 29 10 9 10 057 −129M17
57 38 37 38 029 −157
87 30 14 9 11 157 −329?
56 40 36 36 229 −257?
87 31 22 9 12 257 −529?
55 46 35 34 429 −357?
88 9 3 0 1 155 −232?
78 72 69 70 132 −255?
88 14 8 3 2 332 −255?
73 67 60 63 155 −432?
88 18 12 2 4 255 −432?
69 63 54 54 332 −355?
88 20 5 3 5 077 −210?
67 52 51 51 110 −177?
88 21 6 5 5 133 −154T4
66 51 49 51 054 −233
88 21 9 8 4 511 −176T11 for m=2
66 54 48 54 076 −611T10 for m=2
88 22 6 4 6 076 −211T10 for m=2
65 49 48 48 111 −176T11 for m=2
88 23 7 6 6 132 −155T11 for m=2
64 48 46 48 055 −232T10 for m=2
88 23 17 0 8 176 −911does not exist by N5
does not exist by AbsBd
does not exist by N10
does not exist by N11
64 58 48 42 811 −276does not exist by AbsBd
88 24 15 5 7 254 −433?
63 54 45 45 333 −354?
88 27 12 11 7 510 −177M17
60 45 39 45 077 −610
88 32 12 11 12 055 −132M17
55 35 34 35 032 −155
88 33 27 6 16 177 −1110does not exist by AbsBd
54 48 36 28 1010 −277does not exist by AbsBd
88 36 33 11 17 266 −821?
51 48 31 27 721 −366?
88 37 27 16 15 432 −355?
50 40 27 30 255 −532?
88 39 24 15 19 166 −521?
48 33 27 25 421 −266?
88 40 20 16 20 077 −410T10 for m=2
T12
T18 for (d,l,s)=(4,5,8)
47 27 26 24 310 −177T11 for m=2
T18 for (d,l,s)=(4,5,8)
88 41 23 18 20 155 −332M17
46 28 24 24 232 −255
88 41 38 21 17 722 −365?
46 43 21 27 265 −822?
88 42 21 19 21 066 −221M17
45 24 23 23 121 −166
88 42 24 20 20 233 −254M17
45 27 22 24 154 −333
88 42 27 21 19 422 −265M17
45 30 21 25 165 −522
88 42 36 24 16 1011 −276does not exist by AbsBd
45 39 18 28 176 −1111does not exist by AbsBd
88 43 22 21 21 122 −165T4
T6
44 23 21 23 065 −222T18 for (a,b,r)=(1,1,22)
88 43 23 22 20 311 −176T11 for m=2
44 24 20 24 076 −411T10 for m=2
88 43 32 31 11 212 −185T11 for m=11
44 33 11 33 085 −222T10 for m=11
90 9 1 0 1 080 −19T1
T8(i) for 2-(10,2,1)
T12
T18 for (d,l,s)=(1,1,9)
80 72 71 72 09 −180T8 (ii) for 2-(10,9,8)
T18 for (d,l,s)=(8,8,9)
90 13 4 1 2 155 −234?
76 67 64 65 134 −255?
90 14 4 3 2 225 −164M17
75 65 62 65 064 −325
90 15 3 0 3 084 −35T8(i) for 2-(6,2,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,1,15)
74 62 61 60 25 −184T8 (ii) for 2-(6,5,12)
T11 for m=3
T18 for (d,l,s)=(12,4,15)
90 16 5 2 3 154 −235?
73 62 59 60 135 −254?
90 17 5 4 3 224 −165T11 for m=3
72 60 57 60 065 −324T10 for m=3
90 17 9 8 2 79 −180T8 (ii) for 2-(10,2,1)
T11 for m=2
T18 for (d,l,s)=(1,1,9)
72 64 56 64 080 −89T8(i) for 2-(10,9,8)
T10 for m=2
T12
T18 for (d,l,s)=(8,8,9)
90 18 4 2 4 080 −29T8(i) for 2-(10,3,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,2,9)
71 57 56 56 19 −180T11 for m=2
T18 for (d,l,s)=(7,7,9)
90 19 5 4 4 135 −154T11 for m=2
T20
70 56 54 56 054 −235T10 for m=2
90 22 17 6 5 435 −354?
67 62 49 52 254 −535?
90 24 8 2 8 085 −64does not exist by N9
65 49 48 44 54 −185
90 25 13 4 8 170 −519?
64 52 46 44 419 −270?
90 26 12 11 6 69 −180T8 (ii) for 2-(10,3,2)
T18 for (d,l,s)=(2,2,9)
63 49 42 49 080 −79T12
T18 for (d,l,s)=(7,7,9)
90 26 16 6 8 255 −434M15
63 53 44 44 334 −355
90 27 9 6 9 080 −39T10 for m=3
T12
T18 for (d,l,s)=(3,3,9)
62 44 43 42 29 −180T11 for m=3
T18 for (d,l,s)=(6,6,9)
90 27 17 8 8 340 −349M17
62 52 42 44 249 −440
90 28 11 8 9 150 −239M17
61 44 41 42 139 −250
90 28 16 10 8 425 −264M10
61 49 40 44 164 −525
90 29 11 10 9 220 −169T11 for m=3
60 42 39 42 069 −320T10 for m=3
T18 for (a,b,r)=(2,1,10)
90 29 13 12 8 510 −179T11 for m=2
60 44 38 44 079 −610T10 for m=2
90 29 17 16 6 115 −184T8 (ii) for 2-(6,2,3)
T11 for m=2
T18 for (d,l,s)=(3,1,15)
60 48 36 48 084 −125T8(i) for 2-(6,5,12)
T10 for m=2
T12
T18 for (d,l,s)=(12,4,15)
90 29 19 18 5 144 −185T11 for m=5
60 50 35 50 085 −154T10 for m=5
90 30 11 8 11 079 −310T18 for (a,b,r)=(1,2,5)
59 40 39 38 210 −179
90 30 12 6 12 084 −65T8(i) for 2-(6,3,6)
T10 for m=2
T12
T18 for (d,l,s)=(6,2,15)
59 41 40 36 55 −184T11 for m=2
T18 for (d,l,s)=(9,3,15)
90 30 15 0 15 087 −152T8(i) for 2-(3,2,15)
T10 for m=3
T12
T18 for (d,l,s)=(15,1,30)
59 44 43 30 142 −187T8 (ii) for 2-(3,2,15)
T11 for m=3
T18 for (d,l,s)=(15,1,30)
90 31 13 10 11 149 −240T20
58 40 37 38 140 −249
90 31 17 8 12 169 −520?
58 44 38 36 420 −269?
90 31 25 4 14 179 −1110does not exist by AbsBd
58 52 40 32 1010 −279does not exist by AbsBd
90 32 13 12 11 219 −170M17
57 38 35 38 070 −319
90 32 20 10 12 254 −435M15
57 45 36 36 335 −354
90 33 21 12 12 339 −350M17
56 44 34 36 250 −439
90 34 20 14 12 424 −265?
55 41 32 36 165 −524?
90 35 14 13 14 054 −135M17
54 33 32 33 035 −154
90 35 17 16 12 59 −180T11 for m=2
T18 for (d,l,s)=(3,3,9)
54 36 30 36 080 −69T10 for m=2
T12
T18 for (d,l,s)=(6,6,9)
90 36 16 12 16 080 −49T8(i) for 2-(10,5,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,4,9)
53 33 32 30 39 −180T8 (ii) for 2-(10,6,5)
T11 for m=2
T18 for (d,l,s)=(5,5,9)
90 36 18 9 18 085 −94T8(i) for 2-(5,3,9)
T10 for m=3
T12
T18 for (d,l,s)=(9,2,18)
53 35 34 27 84 −185T8 (ii) for 2-(5,3,9)
T11 for m=3
T18 for (d,l,s)=(9,2,18)
90 37 26 11 18 175 −814?
52 41 32 27 714 −275?
90 38 20 16 16 235 −254?
51 33 28 30 154 −335?
90 38 32 13 18 265 −724?
51 45 30 27 624 −365?
90 39 36 15 18 355 −634?
50 47 28 27 534 −455?
90 40 20 15 20 081 −58T8(i) for 2-(9,5,5)
T10 for m=5
T12
T18 for (d,l,s)=(5,4,10)
49 29 28 25 48 −181T8 (ii) for 2-(9,5,5)
T11 for m=5
T18 for (d,l,s)=(5,4,10)
90 40 38 17 18 445 −544?
49 47 26 27 444 −545?
90 41 24 17 20 163 −426?
48 31 26 25 326 −263?
90 41 25 24 14 114 −185
48 32 20 32 085 −124does not exist by N9
90 41 31 18 19 345 −444?
48 38 25 26 344 −445?
90 41 38 19 18 535 −454?
48 45 24 27 354 −635?
90 42 21 18 21 075 −314T10 for m=3
T12
T18 for (d,l,s)=(3,7,6)
47 26 25 24 214 −175T11 for m=3
T18 for (d,l,s)=(3,7,6)
90 42 26 19 20 245 −344?
47 31 24 25 244 −345?
90 42 36 21 18 625 −364?
47 41 22 27 264 −725?
90 43 23 20 21 145 −244T20
46 26 23 24 144 −245
90 43 26 21 20 327 −262M17
46 29 22 25 162 −427
90 43 32 23 18 715 −274?
46 35 20 27 174 −815?
90 44 22 21 22 045 −144T5
T12
T18 for (d,l,s)=(1,22,2)
T20
45 23 22 23 044 −145T18 for (d,l,s)=(1,22,2)
T20
90 44 23 22 21 215 −174T11 for m=3
45 24 21 24 074 −315T10 for m=3
90 44 24 23 20 49 −180T8 (ii) for 2-(10,5,4)
T11 for m=5
T18 for (d,l,s)=(4,4,9)
45 25 20 25 080 −59T8(i) for 2-(10,6,5)
T10 for m=5
T12
T18 for (d,l,s)=(5,5,9)
90 44 26 25 18 85 −184T8 (ii) for 2-(6,3,6)
T11 for m=3
T18 for (d,l,s)=(6,2,15)
45 27 18 27 084 −95T10 for m=3
T12
T18 for (d,l,s)=(9,3,15)
90 44 29 28 15 143 −186T11 for m=3
45 30 15 30 086 −153T10 for m=3

Table 91-100

v k t λ µ rf sg comments
91 12 6 5 1 513 −177?
78 72 66 72 077 −613?
91 13 2 1 2 077 −113T24
77 66 65 66 013 −177
91 25 10 9 6 413 −177M17
65 50 45 50 077 −513
91 26 8 6 8 077 −213M17
64 46 45 45 113 −177
91 27 18 3 10 177 −813?
63 54 45 40 713 −277?
91 37 30 19 12 913 −277?
53 46 27 36 177 −1013?
91 38 18 17 15 313 −177M17
52 32 28 32 077 −413
91 39 18 15 18 077 −313M17
51 30 29 28 213 −177
91 40 30 13 21 177 −913?
50 40 30 24 813 −277?
92 21 14 7 4 523 −268?
70 63 52 57 168 −623?
92 22 7 6 5 223 −168M17
69 54 51 54 068 −323
92 23 6 5 6 068 −123M17
68 51 50 51 023 −168
92 24 11 4 7 168 −423?
67 54 49 48 323 −268?
92 25 22 3 8 268 −723?
66 63 48 45 623 −368?
92 38 34 12 18 269 −822?
53 49 32 28 722 −369?
92 41 25 16 20 169 −522?
50 34 28 26 422 −269?
92 43 39 22 18 723 −368?
48 44 22 28 268 −823?
92 44 22 20 22 069 −222T10 for m=2
T12
T18 for (d,l,s)=(2,11,4)
47 25 24 24 122 −169T11 for m=2
T18 for (d,l,s)=(2,11,4)
92 44 28 22 20 423 −268M17
47 31 22 26 168 −523
92 45 23 22 22 123 −168T4
T6
T11 for m=2
46 24 22 24 068 −223T10 for m=2
T18 for (a,b,r)=(1,1,23)
93 26 18 5 8 262 −530?
66 58 47 46 430 −362?
93 28 12 7 9 162 −330?
64 48 44 44 230 −262?
93 30 10 9 10 062 −130T12
T18 for (d,l,s)=(1,10,3)
62 42 41 42 030 −162T18 for (d,l,s)=(2,20,3)
93 32 12 11 11 130 −162T18 for (d,l,s)=(1,10,3)
60 40 38 40 062 −230T12
T18 for (d,l,s)=(2,20,3)
93 34 18 13 12 330 −262M17
58 42 35 38 162 −430
93 36 28 15 13 530 −362?
56 48 32 36 262 −630?
94 42 39 18 19 447 −546?
51 48 27 28 446 −547?
94 43 32 19 20 347 −446?
50 39 26 27 346 −447?
94 44 27 20 21 247 −346?
49 32 25 26 246 −347?
94 45 24 21 22 147 −246?
48 27 24 25 146 −247?
94 46 23 22 23 047 −146T5
T12
T18 for (d,l,s)=(1,23,2)
47 24 23 24 046 −147T18 for (d,l,s)=(1,23,2)
T20
95 18 6 5 3 319 −175M17
76 64 60 64 075 −419
95 19 4 3 4 075 −119M17
75 60 59 60 019 −175
95 20 10 1 5 175 −519?
74 64 58 56 419 −275?
95 36 24 16 12 619 −275?
58 46 33 39 175 −719?
95 37 16 15 14 219 −175M17
57 36 33 36 075 −319
95 38 16 14 16 075 −219M17
56 34 33 33 119 −175
95 39 24 13 18 175 −619?
55 40 33 30 519 −275?
96 10 5 1 1 245 −250?
85 80 75 77 150 −345?
96 11 4 3 1 321 −174M17
84 77 73 77 074 −421
96 13 5 0 2 168 −327M11
82 74 70 70 227 −268
96 14 6 2 2 244 −251?
81 73 68 70 151 −344?
96 15 5 4 2 320 −175T11 for m=2
80 70 66 70 075 −420T10 for m=2
96 15 10 9 1 98 −187?
80 75 65 75 087 −108?
96 16 3 1 3 087 −28M17
79 66 65 65 18 −187
96 17 4 3 3 139 −156M17
78 65 63 65 056 −239
96 18 16 0 4 269 −626does not exist by N5
77 75 62 60 526 −369
96 20 5 1 5 090 −45does not exist by N13
75 60 59 57 35 −190
96 21 8 3 5 166 −329?
74 61 57 57 229 −266?
96 21 19 6 4 533 −362?
74 72 56 60 262 −633?
96 22 9 5 5 242 −253M17
73 60 55 57 153 −342
96 22 16 8 4 621 −274?
73 67 54 60 174 −721?
96 23 8 7 5 318 −177T15
72 57 53 57 077 −418T18 for (a,b,r)=(3,1,6)
96 23 11 10 4 79 −186T11 for m=2
72 60 52 60 086 −89T10 for m=2
96 23 14 13 3 116 −189T11 for m=3
72 63 51 63 089 −126T10 for m=3
96 24 7 3 7 089 −46T17
T18 for (a,b,r)=(1,3,2)
71 54 53 51 36 −189
96 24 8 0 8 092 −83T8(i) for 2-(4,2,8)
T10 for m=2
T12
T18 for (d,l,s)=(8,1,24)
71 55 54 48 73 −192T8 (ii) for 2-(4,3,16)
T11 for m=2
T18 for (d,l,s)=(16,2,24)
96 25 10 5 7 165 −330M17
70 55 51 51 230 −265
96 25 15 2 8 180 −715?
70 60 52 48 615 −280?
96 26 11 7 7 241 −254M17
69 54 49 51 154 −341
96 26 20 4 8 268 −627?
69 63 50 48 527 −368?
96 27 10 9 7 317 −178M17
68 51 47 51 078 −417
96 27 23 6 8 356 −539?
68 64 48 48 439 −456?
96 28 24 8 8 444 −451?
67 63 46 48 351 −544?
96 29 23 10 8 532 −363?
66 60 44 48 263 −632?
96 30 10 8 10 080 −215T8(i) for 2-(16,6,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,5,6)
65 45 44 44 115 −180T11 for m=2
T18 for (d,l,s)=(4,10,6)
96 30 15 10 9 332 −263?
65 50 43 46 163 −432?
96 30 20 12 8 620 −275?
65 55 42 48 175 −720?
96 31 11 10 10 132 −163T4
T11 for m=2
64 44 42 44 063 −232T10 for m=2
96 31 15 14 8 78 −187T11 for m=2
64 48 40 48 087 −88T10 for m=2
96 32 11 10 11 063 −132T20
T20
63 42 41 42 032 −163
96 32 12 8 12 087 −48T10 for m=2
63 43 42 40 38 −187T11 for m=2
96 32 16 0 16 093 −162T8(i) for 2-(3,2,16)
T10 for m=2
T12
T18 for (d,l,s)=(16,1,32)
63 47 46 32 152 −193T8 (ii) for 2-(3,2,16)
T11 for m=2
T18 for (d,l,s)=(16,1,32)
96 33 15 10 12 163 −332T20
62 44 40 40 232 −263
96 34 16 12 12 239 −256T20
61 43 38 40 156 −339
96 34 23 10 13 263 −532?
61 50 39 38 432 −363?
96 34 29 17 9 1013 −282does not exist by AbsBd
61 56 35 45 182 −1113does not exist by AbsBd
96 35 15 14 12 315 −180T8 (ii) for 2-(16,6,2)
T11 for m=2
T18 for (d,l,s)=(2,5,6)
60 40 36 40 080 −415T10 for m=2
T12
T18 for (d,l,s)=(4,10,6)
M2 from srg(16,6,2)
96 35 20 19 9 115 −190
60 45 33 45 090 −125does not exist by N13
96 35 25 24 6 193 −192
60 50 30 50 092 −203does not exist by N4
96 36 15 11 15 086 −49M17
59 38 37 35 39 −186
96 36 18 6 18 092 −123does not exist by N4
59 41 40 30 113 −192
96 37 18 13 15 162 −333M17
58 39 35 35 233 −262
96 37 29 8 18 184 −1111does not exist by AbsBd
58 50 38 30 1011 −284does not exist by AbsBd
96 38 19 15 15 238 −257M17
57 38 33 35 157 −338
96 39 18 17 15 314 −181M17
56 35 31 35 081 −414
96 40 20 12 20 090 −85does not exist by N13
55 35 34 28 75 −190
96 40 25 5 25 093 −202does not exist by N3
55 40 39 21 192 −193
96 40 30 20 14 815 −280?
55 45 28 36 180 −915?
96 40 35 13 19 272 −823?
55 50 33 29 723 −372?
96 41 27 14 20 178 −717?
54 40 32 28 617 −278?
96 42 21 15 21 088 −67T10 for m=3
M17
53 32 31 27 57 −188T11 for m=3
96 42 32 16 20 266 −629?
53 43 30 28 529 −366?
96 43 26 17 21 172 −523M17
52 35 29 27 423 −272
96 43 35 18 20 354 −541
52 44 28 28 441 −454M8
96 44 22 18 22 084 −411T10 for m=2
M17
51 29 28 26 311 −184T11 for m=2
96 44 29 19 21 256 −439M17
51 36 27 27 339 −356
96 44 36 20 20 442 −453M8
51 43 26 28 353 −542
96 45 25 20 22 160 −335
50 30 26 26 235 −260M8
96 45 30 21 21 340 −355M17
50 35 25 27 255 −440
96 45 35 22 20 530 −365?
50 40 24 28 265 −630?
96 45 40 23 19 724 −371?
50 45 23 29 271 −824?
96 46 23 21 23 072 −223M17
49 26 25 25 123 −172
96 46 26 22 22 236 −259M8
49 29 24 26 159 −336
96 46 29 23 21 424 −271M17
49 32 23 27 171 −524
96 46 32 24 20 618 −277?
49 35 22 28 177 −718?
96 46 38 26 18 1012 −283does not exist by AbsBd
49 41 20 30 183 −1112does not exist by AbsBd
96 47 24 23 23 124 −171T4
T6
48 25 23 25 071 −224T18 for (a,b,r)=(1,1,24)
96 47 25 24 22 312 −183T11 for m=2
48 26 22 26 083 −412T10 for m=2
M7
96 47 26 25 21 58 −187T11 for m=3
48 27 21 27 087 −68T10 for m=3
96 47 27 26 20 76 −189T11 for m=2
48 28 20 28 089 −86T10 for m=2
M7
96 47 29 28 18 114 −191T11 for m=2
48 30 18 30 091 −124T10 for m=2
96 47 31 30 16 153 −192T8 (ii) for 2-(4,2,8)
T11 for m=2
T18 for (d,l,s)=(8,1,24)
48 32 16 32 092 −163T8(i) for 2-(4,3,16)
T10 for m=2
T12
T18 for (d,l,s)=(16,2,24)
M7
96 47 35 34 12 232 −193T11 for m=2
48 36 12 36 093 −242T10 for m=2
M7
98 13 7 6 1 612 −185T7 for GQ(1,6)
T19
84 78 71 78 085 −712T18 for (a,b,r)=(6,1,2)
98 16 13 0 3 267 −530does not exist by N6
81 78 67 66 430 −367
98 17 15 2 3 353 −444?
80 78 65 66 344 −453?
98 18 15 4 3 439 −358?
79 76 63 66 258 −539?
98 19 13 6 3 525 −272T16
T19
78 72 61 66 172 −625
98 20 9 8 3 611 −186?
77 66 59 66 086 −711?
98 22 12 1 6 180 −617?
75 65 58 55 517 −280?
98 23 16 3 6 266 −531?
74 67 56 55 431 −366?
98 24 18 5 6 352 −445?
73 67 54 55 345 −452?
98 25 18 7 6 438 −359?
72 65 52 55 259 −538?
98 26 16 9 6 524 −273T21
71 61 50 55 173 −624
98 27 12 11 6 610 −187M17
70 55 48 55 087 −710
98 27 17 16 4 135 −192
70 60 46 60 092 −145does not exist by N13
98 28 10 3 10 093 −74does not exist by N9
69 51 50 45 64 −193
98 29 16 5 10 179 −618?
68 55 48 45 518 −279?
98 30 20 7 10 265 −532?
67 57 46 45 432 −365?
98 31 22 9 10 351 −446?
66 57 44 45 346 −451?
98 32 22 11 10 437 −360T21
65 55 42 45 260 −537
98 33 20 13 10 523 −274?
64 51 40 45 174 −623?
98 34 16 15 10 69 −188M17
63 45 38 45 088 −79
98 35 15 8 15 092 −75does not exist by N13
62 42 41 36 65 −192
98 36 21 10 15 178 −619?
61 46 39 36 519 −278?
98 37 25 12 15 264 −533?
60 48 37 36 433 −364?
98 38 27 14 15 350 −447?
59 48 35 36 347 −450?
98 39 27 16 15 436 −361T21
58 46 33 36 261 −536
98 40 25 18 15 522 −275?
57 42 31 36 175 −622?
98 41 21 20 15 68 −189M17
56 36 29 36 089 −78
98 41 25 24 12 134 −193
56 40 26 40 093 −144does not exist by N9
98 42 21 14 21 091 −76T10 for m=7
T12
T18 for (d,l,s)=(7,3,14)
55 34 33 28 66 −191T11 for m=7
T18 for (d,l,s)=(7,3,14)
98 42 24 10 24 094 −143does not exist by N4
55 37 36 24 133 −194
98 42 27 6 27 095 −212does not exist by N3
55 40 39 20 202 −195
98 43 27 16 21 177 −620?
54 38 31 28 520 −277?
98 43 37 12 24 187 −1310does not exist by AbsBd
54 48 34 24 1210 −287does not exist by AbsBd
98 44 31 18 21 263 −534?
53 40 29 28 434 −363?
98 44 40 19 20 449 −548?
53 49 28 29 448 −549?
98 45 33 20 21 349 −448M4
52 40 27 28 348 −449
98 46 28 21 22 249 −348?
51 33 26 27 248 −349?
98 46 33 22 21 435 −362?
51 38 25 28 262 −535?
98 47 25 22 23 149 −248?
50 28 25 26 148 −249?
98 47 31 24 21 521 −276?
50 34 23 28 176 −621?
98 48 24 23 24 049 −148T3
T5
T12
T18 for (d,l,s)=(1,24,2)
49 25 24 25 048 −149T18 for (d,l,s)=(1,24,2)
T20
98 48 27 26 21 67 −190T11 for m=7
49 28 21 28 090 −77T10 for m=7
99 10 2 1 1 144 −154T2
T4
88 80 78 80 054 −244
99 20 8 4 4 244 −254?
78 66 61 63 154 −344?
99 24 12 5 6 254 −344?
74 62 55 56 244 −354?
99 28 14 5 9 177 −521?
70 56 50 48 421 −277?
99 30 10 7 10 088 −310M17
68 48 47 46 210 −188
99 30 18 9 9 344 −354?
68 56 46 48 254 −444?
99 31 12 9 10 155 −243M17
67 48 45 46 143 −255
99 32 12 11 10 222 −176
66 46 43 46 076 −322T18 for (a,b,r)=(2,1,11)
99 32 14 13 9 511 −187T11 for m=3
66 48 42 48 087 −611T10 for m=3
99 33 12 9 12 087 −311T10 for m=3
T25
65 44 43 42 211 −187T11 for m=3
99 34 14 11 12 154 −244M13
64 44 41 42 144 −254
99 34 18 9 13 176 −522?
64 48 42 40 422 −276?
99 34 26 5 15 187 −1111does not exist by AbsBd
64 56 44 36 1011 −287does not exist by AbsBd
99 35 14 13 12 221 −177T11 for m=3
63 42 39 42 077 −321T10 for m=3
99 36 22 13 13 343 −355?
62 48 38 40 255 −443?
99 38 18 17 13 510 −188M17
60 40 34 40 088 −610
99 40 32 16 16 444 −454?
58 50 33 35 354 −544?
99 44 20 19 20 054 −144M17
54 30 29 30 044 −154
100 16 7 0 3 176 −423?
83 74 69 68 323 −276?
100 17 9 2 3 256 −343?
82 74 67 68 243 −356?
100 18 9 4 3 336 −263?
81 72 65 68 163 −436?
100 19 7 6 3 416 −183T9 for pg(5,4,3)
T19
80 68 63 68 083 −516T18 for (a,b,r)=(4,1,4)
100 19 11 10 2 98 −191T11 for m=2
80 72 62 72 091 −108T10 for m=2
100 20 5 0 5 095 −54T8(i) for 2-(5,2,5)
T10 for m=5
T12
T17
T18 for (d,l,s)=(5,1,20)
79 64 63 60 44 −195T8 (ii) for 2-(5,4,15)
T11 for m=5
T18 for (d,l,s)=(15,3,20)
100 21 9 2 5 175 −424?
78 66 61 60 324 −275?
100 22 11 4 5 255 −344?
77 66 59 60 244 −355?
100 23 11 6 5 335 −264T19
76 64 57 60 164 −435
100 24 6 5 6 075 −124T12
T18 for (d,l,s)=(1,6,4)
T20
75 57 56 57 024 −175T18 for (d,l,s)=(3,18,4)
100 24 9 8 5 415 −184T11 for m=5
75 60 55 60 084 −515T10 for m=5
100 27 9 8 7 224 −175T18 for (d,l,s)=(1,6,4)
72 54 51 54 075 −324T12
T18 for (d,l,s)=(3,18,4)
100 28 22 12 6 817 −282?
71 65 48 56 182 −917?
100 29 15 14 6 97 −192?
70 56 46 56 092 −107?
100 30 12 2 12 096 −103does not exist by N4
69 51 50 42 93 −196
100 30 18 11 8 524 −275?
69 57 46 51 175 −624?
100 31 21 4 12 186 −913does not exist by AbsBd
68 58 48 42 813 −286does not exist by AbsBd
100 32 28 6 12 276 −823?
67 63 46 42 723 −376?
100 35 14 9 14 092 −57?
64 43 42 39 47 −192?
100 36 18 11 14 172 −427M17
63 45 40 39 327 −272
100 37 20 13 14 252 −347?
62 45 38 39 247 −352?
100 37 33 16 12 726 −373?
62 58 36 42 273 −826?
100 38 20 15 14 332 −267T21
61 43 36 39 167 −432
100 38 28 18 12 816 −283?
61 51 34 42 183 −916?
100 39 18 17 14 412 −187M17
60 39 34 39 087 −512
100 39 21 20 12 96 −193T11 for m=2
M17
60 42 32 42 093 −106T10 for m=2
100 39 24 23 10 144 −195T8 (ii) for 2-(5,2,5)
T11 for m=5
T18 for (d,l,s)=(5,1,20)
60 45 30 45 095 −154T8(i) for 2-(5,4,15)
T10 for m=5
T12
T18 for (d,l,s)=(15,3,20)
100 39 27 26 8 193 −196
60 48 28 48 096 −203does not exist by N4
100 40 18 13 18 091 −58T18 for (a,b,r)=(2,2,2)
59 37 36 33 48 −191
100 40 20 10 20 095 −104T10 for m=2
T12
T18 for (d,l,s)=(10,2,20)
59 39 38 30 94 −195T11 for m=2
T18 for (d,l,s)=(10,2,20)
100 40 24 4 24 097 −202does not exist by N3
59 43 42 24 192 −197
100 41 22 15 18 171 −428?
58 39 34 33 328 −271?
100 41 29 12 20 185 −914?
58 46 36 30 814 −285?
100 42 24 17 18 251 −348T21
57 39 32 33 248 −351
100 42 36 14 20 275 −824?
57 51 34 30 724 −375?
100 43 24 19 18 331 −268?
56 37 30 33 168 −431?
100 43 41 16 20 365 −734?
56 54 32 30 634 −465?
100 44 22 21 18 411 −188M17
55 33 28 33 088 −511
100 45 27 12 27 096 −153does not exist by N4
54 36 35 22 143 −196
100 45 27 18 22 175 −524?
54 36 30 28 424 −275?
100 46 44 22 20 635 −464?
53 51 26 30 364 −735?
100 47 41 24 20 725 −374?
52 46 24 30 274 −825?
100 48 24 22 24 075 −224T10 for m=2
T12
T18 for (d,l,s)=(2,12,4)
51 27 26 26 124 −175T11 for m=2
T18 for (d,l,s)=(2,12,4)
100 48 30 24 22 425 −274?
51 33 24 28 174 −525?
100 48 36 26 20 815 −284?
51 39 22 30 184 −915?
100 49 25 24 24 125 −174T4
T6
T11 for m=2
50 26 24 26 074 −225T10 for m=2
T18 for (a,b,r)=(1,1,25)
M7
100 49 29 28 20 95 −194T11 for m=2
50 30 20 30 094 −105T10 for m=2
M7

Table 101-110

v k t λ µ rf sg comments
102 16 6 5 2 417 −184?
85 75 70 75 084 −517?
102 17 3 2 3 084 −117M17
84 70 69 70 017 −184
102 29 19 6 9 268 −533?
72 62 51 50 433 −368?
102 31 13 8 10 168 −333?
70 52 48 48 233 −268?
102 32 24 14 8 817 −284?
69 61 44 52 184 −917?
102 33 11 10 11 068 −133T12
T18 for (d,l,s)=(1,11,3)
68 46 45 46 033 −168T18 for (d,l,s)=(2,22,3)
102 33 13 12 10 317 −184T11 for m=2
68 48 44 48 084 −417T10 for m=2
102 34 12 10 12 084 −217T10 for m=2
67 45 44 44 117 −184T11 for m=2
102 34 17 0 17 099 −172T8(i) for 2-(3,2,17)
T10 for m=17
T12
T18 for (d,l,s)=(17,1,34)
67 50 49 34 162 −199T8 (ii) for 2-(3,2,17)
T11 for m=17
T18 for (d,l,s)=(17,1,34)
102 35 13 12 12 133 −168T11 for m=2
T18 for (d,l,s)=(1,11,3)
66 44 42 44 068 −233T10 for m=2
T12
T18 for (d,l,s)=(2,22,3)
102 35 21 8 14 184 −717?
66 52 44 40 617 −284?
102 37 19 14 13 333 −268?
64 46 39 42 168 −433?
102 39 29 16 14 533 −368?
62 52 36 40 268 −633?
102 43 29 14 21 185 −816?
58 44 35 30 716 −285?
102 46 41 20 21 451 −550?
55 50 29 30 450 −551?
102 47 34 21 22 351 −450?
54 41 28 29 350 −451?
102 48 24 21 24 085 −316T10 for m=3
T12
T18 for (d,l,s)=(3,8,6)
53 29 28 27 216 −185T11 for m=3
T18 for (d,l,s)=(3,8,6)
102 48 29 22 23 251 −350T20
53 34 27 28 250 −351
102 49 26 23 24 151 −250T20
52 29 26 27 150 −251
102 49 35 26 21 717 −284?
52 38 23 30 184 −817?
102 50 25 24 25 051 −150T5
T12
T18 for (d,l,s)=(1,25,2)
T20
51 26 25 26 050 −151T18 for (d,l,s)=(1,25,2)
T20
102 50 26 25 24 217 −184T11 for m=3
51 27 24 27 084 −317T10 for m=3
102 50 33 32 17 163 −198T11 for m=17
51 34 17 34 098 −173T10 for m=17
104 11 7 2 1 339 −264?
92 88 81 84 164 −439?
104 14 4 1 2 164 −239?
89 79 76 77 139 −264?
104 22 13 3 5 265 −438?
81 72 63 63 338 −365?
104 24 6 4 6 091 −212T8(i) for 2-(13,4,2)
T10 for m=2
T12
T18 for (d,l,s)=(2,3,8)
79 61 60 60 112 −191T11 for m=2
T18 for (d,l,s)=(6,9,8)
104 25 7 6 6 139 −164T4
T11 for m=2
78 60 58 60 064 −239T10 for m=2
104 25 10 9 5 513 −190?
78 63 57 63 090 −613?
104 26 7 5 7 090 −213M17
77 58 57 57 113 −190
104 27 8 7 7 138 −165T20
76 57 55 57 065 −238
104 27 18 1 9 190 −913does not exist by AbsBd
76 67 57 51 813 −290does not exist by AbsBd
104 28 16 6 8 264 −439?
75 63 54 54 339 −364?
104 31 13 12 8 512 −191T8 (ii) for 2-(13,4,2)
T11 for m=2
T18 for (d,l,s)=(2,3,8)
72 54 48 54 091 −612T10 for m=2
T12
T18 for (d,l,s)=(6,9,8)
104 36 24 13 12 439 −364?
67 55 42 45 264 −539?
104 39 15 14 15 064 −139M17
64 40 39 40 039 −164
104 41 31 10 20 191 −1112does not exist by AbsBd
62 52 40 32 1012 −291does not exist by AbsBd
104 42 36 15 18 364 −639?
61 55 36 35 539 −464?
104 44 37 15 21 278 −825?
59 52 35 31 725 −378?
104 47 28 19 23 178 −525?
56 37 31 29 425 −278?
104 48 24 20 24 091 −412T10 for m=2
T12
T18 for (d,l,s)=(4,6,8)
55 31 30 28 312 −191T11 for m=2
T18 for (d,l,s)=(4,6,8)
104 49 27 22 24 165 −338M17
54 32 28 28 238 −265
104 49 42 25 21 726 −377?
54 47 25 31 277 −826?
104 50 25 23 25 078 −225M17
53 28 27 27 125 −178
104 50 28 24 24 239 −264M17
53 31 26 28 164 −339
104 50 31 25 23 426 −277M17
53 34 25 29 177 −526
104 50 40 28 20 1013 −290does not exist by AbsBd
53 43 22 32 190 −1113does not exist by AbsBd
104 51 26 25 25 126 −177T4
T6
52 27 25 27 077 −226T18 for (a,b,r)=(1,1,26)
104 51 27 26 24 313 −190T11 for m=2
52 28 24 28 090 −413T10 for m=2
104 51 38 37 13 252 −1101T11 for m=13
52 39 13 39 0101 −262T10 for m=13
105 14 2 1 2 090 −114T8(i) for 2-(15,3,1)
T12
T18 for (d,l,s)=(1,2,7)
90 78 77 78 014 −190T18 for (d,l,s)=(6,12,7)
105 18 12 3 3 349 −355?
86 80 70 72 255 −449?
105 20 4 3 4 084 −120T8(i) for 2-(21,5,1)
T12
T18 for (d,l,s)=(1,4,5)
84 68 67 68 020 −184T18 for (d,l,s)=(4,16,5)
105 20 8 7 3 514 −190T8 (ii) for 2-(15,3,1)
T11 for m=3
T18 for (d,l,s)=(1,2,7)
84 72 66 72 090 −614T10 for m=3
T12
T18 for (d,l,s)=(6,12,7)
105 24 8 7 5 320 −184T8 (ii) for 2-(21,5,1)
T9 for pg(5,5,5)
T18 for (d,l,s)=(1,4,5)
80 64 60 64 084 −420T12
T18 for (d,l,s)=(4,16,5)
105 28 8 6 8 090 −214T12
T18 for (d,l,s)=(2,4,7)
76 56 55 55 114 −190T18 for (d,l,s)=(5,10,7)
105 28 20 11 6 720 −284?
76 68 53 60 184 −820?
105 30 10 5 10 098 −56T8(i) for 2-(7,3,5)
T10 for m=5
T12
T18 for (d,l,s)=(5,2,15)
74 54 53 50 46 −198T8 (ii) for 2-(7,5,10)
T11 for m=5
T18 for (d,l,s)=(10,4,15)
105 31 14 7 10 177 −427?
73 56 51 50 327 −277?
105 32 16 9 10 256 −348?
72 56 49 50 248 −356?
105 32 24 11 9 535 −369?
72 64 48 52 269 −635?
105 33 16 11 10 335 −269?
71 54 47 50 169 −435?
105 34 12 11 11 135 −169T4
70 48 46 48 069 −235
105 34 14 13 10 414 −190T11 for m=5
T18 for (d,l,s)=(2,4,7)
70 50 45 50 090 −514T10 for m=5
T12
T18 for (d,l,s)=(5,10,7)
105 34 20 19 7 135 −199T11 for m=7
70 56 42 56 099 −145T10 for m=7
105 35 12 11 12 069 −135T20
69 46 45 46 035 −169
105 35 14 7 14 099 −75T10 for m=7
69 48 47 42 65 −199T11 for m=7
105 36 16 11 13 169 −335M10
68 48 44 44 235 −269
105 36 20 9 14 184 −620?
68 52 45 42 520 −284?
105 36 24 7 15 190 −914?
68 56 46 40 814 −290?
105 37 24 11 14 269 −535?
67 54 43 42 435 −369?
105 38 26 13 14 354 −450?
66 54 41 42 350 −454?
105 38 36 11 15 369 −735?
66 64 42 40 635 −469?
105 39 26 15 14 439 −365?
65 52 39 42 265 −539?
105 40 16 14 16 084 −220T12
T18 for (d,l,s)=(2,8,5)
64 40 39 39 120 −184T18 for (d,l,s)=(3,12,5)
105 40 24 17 14 524 −280?
64 48 37 42 180 −624?
105 40 32 20 12 1014 −290?
64 56 35 45 190 −1114?
105 41 20 19 14 69 −195T11 for m=7
63 42 35 42 095 −79T10 for m=7
105 42 18 15 18 090 −314T8(i) for 2-(15,7,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,6,7)
62 38 37 36 214 −190T11 for m=3
T18 for (d,l,s)=(4,8,7)
105 42 36 19 15 727 −377?
62 56 34 40 277 −827?
105 43 20 17 18 155 −249?
61 38 35 36 149 −255?
105 44 20 19 18 220 −184T11 for m=3
T18 for (d,l,s)=(2,8,5)
60 36 33 36 084 −320T10 for m=3
T12
T18 for (d,l,s)=(3,12,5)
105 44 24 23 15 96 −198T8 (ii) for 2-(7,3,5)
T11 for m=5
T18 for (d,l,s)=(5,2,15)
60 40 30 40 098 −106T8(i) for 2-(7,5,10)
T10 for m=5
T12
T18 for (d,l,s)=(10,4,15)
105 48 24 23 21 314 −190T8 (ii) for 2-(15,7,3)
T18 for (d,l,s)=(3,6,7)
56 32 28 32 090 −414T12
T18 for (d,l,s)=(4,8,7)
105 48 32 24 20 620 −284?
56 40 27 33 184 −720?
105 50 38 19 28 190 −1014?
54 42 31 24 914 −290?
106 48 42 21 22 453 −552?
57 51 30 31 452 −553?
106 49 35 22 23 353 −452M17
56 42 29 30 352 −453
106 50 30 23 24 253 −352?
55 35 28 29 252 −353?
106 51 27 24 25 153 −252M17
54 30 27 28 152 −253
106 52 26 25 26 053 −152T3
T5
T12
T18 for (d,l,s)=(1,26,2)
53 27 26 27 052 −153T18 for (d,l,s)=(1,26,2)
T20
108 10 3 0 1 168 −239M11
97 90 87 88 139 −268
108 11 3 2 1 232 −175M16
96 88 85 88 075 −332
108 14 10 0 2 269 −438M16
93 89 80 80 338 −369
108 15 11 2 2 351 −356?
92 88 78 80 256 −451?
108 16 10 4 2 433 −274?
91 85 76 80 174 −533?
108 17 7 6 2 515 −192T11 for m=2
90 80 74 80 092 −615T10 for m=2
T18 for (a,b,r)=(5,1,3)
108 19 9 0 4 186 −521?
88 78 72 70 421 −286?
108 20 12 2 4 268 −439?
87 79 70 70 339 −368?
108 21 13 4 4 350 −357?
86 78 68 70 257 −450?
108 22 12 6 4 432 −275M16
85 75 66 70 175 −532
108 23 9 8 4 514 −193T11 for m=2
M16
84 70 64 70 093 −614T10 for m=2
108 24 6 3 6 099 −38T8(i) for 2-(9,3,3)
T10 for m=3
T12
T18 for (d,l,s)=(3,2,12)
83 65 64 63 28 −199T11 for m=3
T18 for (d,l,s)=(9,6,12)
108 25 8 5 6 163 −244?
82 65 62 63 144 −263?
108 25 15 8 5 527 −280M16
82 72 61 66 180 −627
108 26 8 7 6 227 −180T11 for m=3
81 63 60 63 080 −327T10 for m=3
108 27 7 6 7 080 −127T20
T20
80 60 59 60 027 −180
108 27 9 0 9 0104 −93T8(i) for 2-(4,2,9)
T10 for m=3
T12
T18 for (d,l,s)=(9,1,27)
80 62 61 54 83 −1104T8 (ii) for 2-(4,3,18)
T11 for m=3
T18 for (d,l,s)=(18,2,27)
108 28 12 5 8 180 −427?
79 63 58 57 327 −280?
108 28 17 2 9 192 −815?
79 68 59 54 715 −292?
108 29 23 4 9 280 −727?
78 72 57 54 627 −380?
108 30 10 4 10 0102 −65does not exist by N13
77 57 56 52 55 −1102
108 30 27 6 9 368 −639?
77 74 55 54 539 −468?
108 31 15 6 10 184 −523?
76 60 54 52 423 −284?
108 31 29 8 9 456 −551?
76 74 53 54 451 −556?
108 32 18 8 10 266 −441?
75 61 52 52 341 −366?
108 32 29 10 9 544 −463?
75 72 51 54 363 −644?
108 33 11 8 11 096 −311M17
74 52 51 50 211 −196
108 33 19 10 10 348 −359?
74 60 50 52 259 −448?
108 33 27 12 9 632 −375?
74 68 49 54 275 −732?
108 34 13 10 11 160 −247M17
73 52 49 50 147 −260
108 34 18 12 10 430 −277M16
73 57 48 52 177 −530
108 34 23 14 9 720 −287?
73 62 47 54 187 −820?
108 34 28 16 8 1015 −292?
73 67 46 56 192 −1115?
108 35 13 12 11 224 −183
72 50 47 50 083 −324T18 for (a,b,r)=(2,1,12)
108 35 15 14 10 512 −195T11 for m=2
72 52 46 52 095 −612T10 for m=2
108 35 17 16 9 88 −199T8 (ii) for 2-(9,3,3)
T11 for m=3
T18 for (d,l,s)=(3,2,12)
72 54 45 54 099 −98T10 for m=3
T12
T18 for (d,l,s)=(9,6,12)
108 35 19 18 8 116 −1101T11 for m=2
72 56 44 56 0101 −126T10 for m=2
108 35 23 22 6 174 −1103T11 for m=2
72 60 42 60 0103 −184T10 for m=2
108 36 13 10 13 095 −312T18 for (a,b,r)=(1,2,6)
71 48 47 46 212 −195
108 36 14 8 14 0101 −66T10 for m=2
71 49 48 44 56 −1101T11 for m=2
108 36 15 6 15 0103 −94T10 for m=3
71 50 49 42 84 −1103T11 for m=3
108 36 16 4 16 0104 −123does not exist by N4
71 51 50 40 113 −1104
108 36 18 0 18 0105 −182T8(i) for 2-(3,2,18)
T10 for m=2
T12
T18 for (d,l,s)=(18,1,36)
71 53 52 36 172 −1105T8 (ii) for 2-(3,2,18)
T11 for m=2
T18 for (d,l,s)=(18,1,36)
108 37 15 12 13 159 −248T20
70 48 45 46 148 −259
108 37 19 10 14 183 −524M17
70 52 46 44 424 −283
108 37 23 8 15 191 −816?
70 56 47 42 716 −291?
108 37 27 6 16 195 −1112does not exist by AbsBd
70 60 48 40 1012 −295does not exist by AbsBd
108 38 15 14 13 223 −184M17
69 46 43 46 084 −323
108 38 22 12 14 265 −442M16
69 53 44 44 342 −365
108 38 29 10 15 279 −728?
69 60 45 42 628 −379?
108 39 23 14 14 347 −360M16
68 52 42 44 260 −447
108 39 33 12 15 367 −640?
68 62 43 42 540 −467?
108 40 22 16 14 429 −278?
67 49 40 44 178 −529?
108 40 35 14 15 455 −552?
67 62 41 42 452 −555?
108 41 19 18 14 511 −196T11 for m=2
M17
66 44 38 44 096 −611T10 for m=2
108 41 35 16 15 543 −464M16
66 60 39 42 364 −643
108 42 33 18 15 631 −376M16
65 56 37 42 276 −731
108 43 29 20 15 719 −288?
64 50 35 42 188 −819?
108 44 23 22 15 87 −1100?
63 42 33 42 0100 −97?
108 44 30 29 10 203 −1104
63 49 28 49 0104 −213does not exist by N4
108 45 20 17 20 092 −315M17
62 37 36 35 215 −192
108 45 25 10 25 0104 −153does not exist by N4
62 42 41 28 143 −1104
108 46 22 19 20 156 −251M16
61 37 34 35 151 −256
108 46 36 24 16 1014 −293does not exist by AbsBd
61 51 30 40 193 −1114does not exist by AbsBd
108 46 38 16 22 281 −826?
61 53 36 32 726 −381?
108 47 22 21 20 220 −187?
60 35 32 35 087 −320?
108 47 27 26 16 115 −1102
60 40 28 40 0102 −125does not exist by N13
108 48 24 18 24 099 −68T10 for m=2
T12
T18 for (d,l,s)=(6,4,12)
59 35 34 30 58 −199T11 for m=2
T18 for (d,l,s)=(6,4,12)
108 48 32 8 32 0105 −242does not exist by N3
59 43 42 20 232 −1105
108 49 29 20 24 181 −526T20
58 38 32 30 426 −281
108 50 32 22 24 263 −444T20
57 39 30 30 344 −363
108 51 33 24 24 345 −362T20
56 38 28 30 262 −445
108 51 43 26 22 727 −380?
56 48 26 32 280 −827?
108 52 26 24 26 081 −226T10 for m=2
T12
T18 for (d,l,s)=(2,13,4)
55 29 28 28 126 −181T11 for m=2
T18 for (d,l,s)=(2,13,4)
108 52 32 26 24 427 −280M17
55 35 26 30 180 −527
108 53 27 26 26 127 −180T4
T6
T11 for m=2
54 28 26 28 080 −227T10 for m=2
T18 for (a,b,r)=(1,1,27)
M7
108 53 29 28 24 59 −198T11 for m=2
54 30 24 30 098 −69T10 for m=2
M7
108 53 35 34 18 173 −1104T8 (ii) for 2-(4,2,9)
T11 for m=2
T18 for (d,l,s)=(9,1,27)
54 36 18 36 0104 −183T8(i) for 2-(4,3,18)
T10 for m=2
T12
T18 for (d,l,s)=(18,2,27)
M7
110 10 1 0 1 099 −110T1
T8(i) for 2-(11,2,1)
T12
T18 for (d,l,s)=(1,1,10)
99 90 89 90 010 −199T8 (ii) for 2-(11,10,9)
T18 for (d,l,s)=(9,9,10)
110 19 10 9 2 810 −199T8 (ii) for 2-(11,2,1)
T18 for (d,l,s)=(1,1,10)
90 81 72 81 099 −910T8(i) for 2-(11,10,9)
T12
T18 for (d,l,s)=(9,9,10)
110 19 15 4 3 444 −365?
90 86 73 76 265 −544?
110 20 4 2 4 099 −210T10 for m=2
T12
T18 for (d,l,s)=(2,2,10)
89 73 72 72 110 −199T11 for m=2
T18 for (d,l,s)=(8,8,10)
110 21 5 4 4 144 −165T4
T11 for m=2
88 72 70 72 065 −244T10 for m=2
110 23 7 4 5 165 −244?
86 70 67 68 144 −265?
110 25 21 4 6 365 −544?
84 80 64 64 444 −465?
110 29 13 12 6 710 −199T11 for m=2
T18 for (d,l,s)=(2,2,10)
80 64 56 64 099 −810T10 for m=2
T12
T18 for (d,l,s)=(8,8,10)
110 30 9 6 9 099 −310T12
T18 for (d,l,s)=(3,3,10)
79 58 57 56 210 −199T18 for (d,l,s)=(7,7,10)
110 31 25 0 12 199 −1310does not exist by N5
does not exist by AbsBd
does not exist by N10
does not exist by N11
78 72 58 48 1210 −299does not exist by AbsBd
110 39 18 17 12 610 −199T18 for (d,l,s)=(3,3,10)
70 49 42 49 099 −710T12
T18 for (d,l,s)=(7,7,10)
110 40 16 12 16 099 −410T8(i) for 2-(11,5,4)
T10 for m=2
T12
T18 for (d,l,s)=(4,4,10)
69 45 44 42 310 −199T11 for m=2
T18 for (d,l,s)=(6,6,10)
110 40 34 15 14 544 −465?
69 63 42 45 365 −644?
110 41 34 7 20 199 −1410does not exist by AbsBd
68 61 46 35 1310 −299does not exist by AbsBd
110 42 20 16 16 244 −265?
67 45 40 42 165 −344?
110 44 18 17 18 065 −144M17
65 39 38 39 044 −165
110 44 22 11 22 0105 −114T10 for m=11
T12
T18 for (d,l,s)=(11,2,22)
65 43 42 33 104 −1105T11 for m=11
T18 for (d,l,s)=(11,2,22)
110 45 32 13 22 195 −1014does not exist by AbsBd
64 51 40 33 914 −295does not exist by AbsBd
110 46 28 18 20 265 −444?
63 45 36 36 344 −365?
110 46 40 15 22 285 −924?
63 57 38 33 824 −385?
110 49 25 24 20 510 −199T8 (ii) for 2-(11,5,4)
T11 for m=2
T18 for (d,l,s)=(4,4,10)
60 36 30 36 099 −610T10 for m=2
T12
T18 for (d,l,s)=(6,6,10)
110 50 25 20 25 099 −510T10 for m=5
T12
T18 for (d,l,s)=(5,5,10)
59 34 33 30 410 −199T11 for m=5
T18 for (d,l,s)=(5,5,10)
110 50 43 22 23 455 −554?
59 52 31 32 454 −555?
110 51 29 22 25 177 −432?
58 36 31 30 332 −277?
110 51 36 23 24 355 −454?
58 43 30 31 354 −455?
110 52 31 24 25 255 −354?
57 36 29 30 254 −355?
110 52 46 27 22 825 −384?
57 51 26 33 284 −925?
110 53 28 25 26 155 −254T20
56 31 28 29 154 −255
110 53 31 26 25 333 −276?
56 34 27 30 176 −433?
110 53 40 29 22 915 −294?
56 43 24 33 194 −1015?
110 53 46 31 20 1311 −298does not exist by AbsBd
56 49 22 35 198 −1411does not exist by AbsBd
110 54 27 26 27 055 −154T5
T12
T18 for (d,l,s)=(1,27,2)
55 28 27 28 054 −155T18 for (d,l,s)=(1,27,2)
T20
110 54 29 28 25 411 −198T11 for m=5
55 30 25 30 098 −511T10 for m=5
110 54 32 31 22 105 −1104T11 for m=11
55 33 22 33 0104 −115T10 for m=11

References

[1]

Alexander Araluze, Klavdija Kutnar, Luis Martínez & Dragan Marušič, Edge connectivity in difference graphs and some new constructions of partial sum families, Europ. J. Combin. 32 (2011) 352-360.

[2]

A. E. Brouwer & Š. Gyürki, Enumeration of small vertex-transitive directed strongly regular graphs, arXiv, Oct. 2026.

[3]

V. A. Byzov & I. A. Pushkarev, On the existence of directed strongly regular graphs with parameters (22,9,6,3,4), url, 2024.

[4]

V. A. Byzov & I. A. Pushkarev, On the existence of strongly regular digraphs with parameters (9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4), arXiv, 2026.

[5]

Stephen Chung & William J. Wesley, AI research conducted on rStation, with results available on GitHub, Oct. 2026.

[6]

D. Crnković & A. Švob, The existence of directed strongly regular graphs with parameters (63,11,8,1,2), arXiv, 2024.

[7]

D. Crnković, A. Švob & M. Zubović Žutolija, New directed strongly regular graphs on 60 vertices, arXiv, 2026.

[8]

D. Crnković, A. Švob & M. Zubović Žutolija, The existence of some directed strongly regular graphs on 108 vertices, arXiv, 2026.

[9]

A. Duval, A directed graph version of strongly regular graphs, JCT (A) 47 (1988) 71-100.

[10]

Art M. Duval & Dmitri Iourinski, Semidirect product constructions of directed strongly regular graphs, JCT (A) 104 (2003) 157-167. Preprint, 2000.

[11]

Frank Fiedler, Mikhail H. Klin & Mikhail Muzychuk, Small vertex-transitive directed strongly regular graphs, Discr. Math. 255 (2002) 87-115. (The preprint was called A census of ....)

[12]

F. Fiedler, M. Klin & Ch. Pech, Directed strongly regular graphs as elements of coherent algebras, in: General Algebra and Discrete Mathematics: proceedings of the Conference on General Algebra and Discrete Mathematics, Potsdam 1998, K. Denecke & H.-J. Vogel (eds.), Shaker Verlag, Aachen 1999, 69-87.

[13]

J. Gimbert, On digraphs with unique walks of closed lengths between vertices, Australas. J. Combin. 20 (1999) 77-90.

[14]

C. D. Godsil, S. A. Hobart & W. J. Martin, Representations of directed strongly regular graphs, Europ. J. Comb. 28 (2007) 1980-1993.

[15]

Štefan Gyürki, Infinite families of directed strongly regular graphs using equitable partitions, Discr. Math. 339 (2016) 2970-2986.

[16]

Štefan Gyürki, Small Directed Strongly Regular Graphs, Algebra Colloquium 27 (2020) 11-30.

[17]

Štefan Gyürki, A systematic search for Vertex Transitive Directed Strongly Regular Graphs, pp. 22-27 in: 9th International Symposium on Symbolic Computation in Software Science, 2021. RISC Report Series No. 21-16.

[18]

Štefan Gyürki & Mikhail Klin, Sporadic examples of Directed Strongly Regular Graphs obtained by Computer Algebra experimentation, Aug 2014, arXiv and (without appendix) pp. 155-170 in Computer algebra in scientific computing, Proc. 16th international workshop, CASC 2014, Warsaw, Poland, September 8-12, 2014 (V. P. Gerdt et al., eds.), Springer LNCS 8660.

[19]

Štefan Gyürki & Mikhail Klin, Directed strongly regular graphs from groups, loops and quasigroups, arXiv, 2026.

[20]

J. M. Hammersley, The friendship theorem and the love problem, pp. 31-54 in: Surveys in Combinatorics (E. Keith Lloyd, ed.), Cambridge University Press 1983. [This contains pictures of dsrg(8,3,2,1,1) and dsrg(15,4,2,1,1).]

[21]

S. A. Hobart & T. J. Shaw, A note on a family of directed strongly regular graphs, Europ. J. Combin. 20 (1999) 819-820.

[22]

S. A. Hobart & J. Williford, New feasibility conditions for directed strongly regular graphs, Electr. J. Comb. 24 (2017) #P1,16.

[23]

Allan van Hulst, emails, Jan-Jun 2025.

[24]

Leif K. Jørgensen, Directed strongly regular graphs with μ = λ, Discrete Math. 231 (2001) 289-293. Aalborg University preprint R-99-2009, August 1999.

[25]

Leif K. Jørgensen, Search for directed strongly regular graphs, Aalborg University preprint R-99-2016, October 1999.

[26]

Leif Kjær Jørgensen, Non-existence of directed strongly regular graphs, Discrete Math. 264 (2003) 111-126. Aalborg University preprint R-00-2024, September 2000.

[27]

Leif Kjær Jørgensen, Directed strongly regular graphs with rank 5, Lin. Alg. Appl. 477 (2015) 102-111. Aalborg University preprint R-2014-05, May 2014.

[28]

Leif K. Jørgensen, New mixed Moore graphs and directed strongly regular graphs, Discr. Math. 338 (2015) 1011-1016.

[29]

M. Klin, A. Munemasa, M. Muzychuk & P.-H. Zieschang, Directed strongly regular graphs obtained from coherent algebras, Lin. Alg. Appl. 377 (2004) 83-109. Preprint, Kyushu MPS-1997-12, Kyushu University, 1997. Maybe also preprint Ben Gurion University, 1997.

[30]

Mikhail Klin, Christian Pech & Paul-Hermann Zieschang, Flag algebras of block designs: I. Initial notions, Steiner 2-designs, and generalized quadrangles, preprint MATH-AL-10-1998, Technische Universität Dresden.

[31]

N. López, J. M. Miret & C. Fernández, Non existence of some mixed Moore graphs of diameter 2 using SAT, Discr. Math. 339 (2016) 589-596.

[32]

Luis Martínez & Alexander Araluze, New tools for the construction of directed strongly regular graphs: Difference digraphs and partial sum families, JCT (B) 100 (2010) 720-728.

[33]

Luis Martinez, Strongly regular m-Cayley circulant graphs and digraphs, Ars Math. Contemporanea 8 (2015) 195-213.

[34]

Jerod Michel, A note on directed strongly regular graphs, Graphs Combin. 33 (2017) 171-179. (See also Jerod Michel & Baokun Ding, Notes on various methods for constructing directed strongly regular graphs, arXiv, 2015.)

[35]

A. Neumaier, t½-designs, J. Combin. Th. (A) 28 (1980) 226-248.

[36]

Oktay Olmez & Sung Y. Song, Construction of directed strongly regular graphs using finite incidence structures, preprint, 2010. (See also A. E. Brouwer, O. Olmez & S. Y. Song, Directed strongly regular graphs from 1½-designs, Europ. J. Comb. 33 (2012) 1174-1177.)

[37]

Oktay Olmez & Sung Y. Song, Some families of directed strongly regular graphs obtained from certain finite incidence structures, Graphs and Combinatorics 30 (2014) 1529-1549. (See also arXiv.)

[38]

Jingkun Zhou, Zhiwen He & Zhao Chai, Two kinds of constructions of directed strongly regular graphs from partial sum families and semi-direct products of groups, arXiv, Feb 2020.