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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 [10].)

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

Hammersley [10] found a dsrg(15,4,2,1,1) and is inclined to believe that it is the only one. However, Jørgensen [14] 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 [10] 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. [18] 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. [4] 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 [6] showed that dsrg(k(k+1),k,1,0,1) is unique, and Jørgensen [15] 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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