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3. 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 [11].)

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

Hammersley [11] found a dsrg(15,4,2,1,1) and is inclined to believe that it is the only one. However, Jørgensen [15] 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 [11]
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. [19]
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. [5]
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 [7] showed that dsrg(k(k+1),k,1,0,1) is unique, and Jørgensen [16] extends this to: dsrg(m(m+1)t,mt, t,0,t) is unique for all positive integers m and t.


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