## Steiner 2-designs

### Designs S(2,4,28) with nontrivial automorphisms

There are 4466 designs S(2,4,28) with nontrivial automorphism groups. They were classified in the paper Glas. Mat. Ser. III 37(57) (2002), 259-268. A gnu-zipped list of incidence matrices is available for download. The following table contains the distribution of the designs by order of full automorphism group.

 |Aut| Freq. |Aut| Freq. |Aut| Freq. |Aut| Freq. 12096 1 48 12 18 1 6 60 1512 1 42 1 16 10 4 374 216 1 32 2 12 12 3 1849 192 2 27 1 9 18 2 2028 72 1 24 12 8 71 64 1 21 6 7 2

Designs with these parameters are unitals of order 3. Exactly 17 of them can be embedded in projective planes of order 9. The embedded S(2,4,28)s were classified by Pentilla and Royle in Des. Codes Cryptography 6 (1995), 229-245. We repeated the calculation and got the same numbes of inequivalent unitals:

One of the unitals in the Hall plane is self-dual, so the total number of embeddable S(2,4,28)s is 17 up to isomorphism. They are designs number 381, 503, 520, 521, 589, 594, 597, 711, 713, 912, 913, 944, 1000, 1002, 1050, 1171, and 4460 in our list of incidence matrices.

### Designs S(2,4,37)

There are two S(2,4,37) designs with automorphisms of order 37 and 284 with automorphisms of order 11. In the paper Ars Combin. 78 (2006), 127-135 automorphisms of order 2 and 3 were studied and used to find many more examples. Here is a list of 51402 non-isomorphic S(2,4,37) designs. Some of them contain S(2,3,9) subdesigns, closing a gap in the embedding spectrum of S(2,3,9) into S(2,4,v) (see M.Meszka, A.Rosa, Embedding Steiner triple systems into Steiner systems S(2,4,v), Discrete Math. 274 (2004), 199-212). Below is the distribution by full automorphism group order.

 |Aut| Freq. 111 1 54 4 37 1 33 4 27 2 18 7 11 280 9 203 3 1748 2 49152

Involutory automorphisms with the maximum number of fixed points proved particularly prolific. The corresponding orbit matrices contain a linear space with 13 points and 23 lines as the fixed part, and (12,3,2) BIBDs as the non-fixed part. Here are 5000 such orbit matrices. They can be indexed to more than 12 million incidence matrices of S(2,4,37) designs. Most of them are probably non-isomorphic, but this has been verified only for designs arising from a dozen of orbit matrices (these are the 49152 designs with |Aut|=2).

### Designs S(2,5,41)

R.Mathon and A.Rosa classified S(2,5,41) designs with automorphisms of order 5. Four designs were found and another one (with full automorphism group of order 24) was obtained by applying a transformation. I managed to find all S(2,5,41)s with automorphisms of order 3. There are 12 such designs, nine of which were previously unknown. This result was published in J. Combin. Math. Combin. Comput. 43 (2002), 83-99. Subsequently I also classified S(2,5,41)s with automorphisms of order 4, but all such designs were already known. However, the search produced a new design with a single involution. The result was presented at the 2nd Croatian Mathematical Congress.

Thus, there are at least 15 non-isomorphic S(2,5,41) designs. Here is a list of incidence matrices, and the following table contains distribution by size of full automorphism group.

 |Aut| Freq. 205 1 120 2 24 2 20 1 18 4 12 1 9 1 6 2 2 1

### Designs S(2,5,45)

There are exactly three S(2,5,45) designs with automorphisms of order 5 (full automorphism groups are of order 360, 160 and 40). The incidence matrices can be downloaded here. The three designs are not resolvable. As far as I know existence of a resolvable S(2,5,45) is still in question.

### Designs S(2,5,65)

Designs with parameters S(2,5,65) are unitals of order 4. Stoichev and Tonchev performed a nonexhaustive search for unitals in projective planes of order 16 in Discrete Appl. Math. 102 (2000), 151-158. From their data 73 non-isomorphic S(2,5,65) designs can be reconstructed. Two further ones with cyclic automorphism groups are known. Together with A. Nakic and M.O. Pavcevic, we classified the S(2,5,65) designs with a nonabelian group of order 39 (there are 1284), and costructed more examples with other automorphism groups. The work is described in a paper submitted for publication.

The total number of non-isomorphic S(2,5,65)s is at least 1777. A gnu-zipped list of incidence matrices is available here, and the following table contains the distribution by full automorphism group size.

 |Aut| Freq. |Aut| Freq. |Aut| Freq. |Aut| Freq. |Aut| Freq. 249600 1 300 10 150 2 64 67 20 2 1200 1 260 1 128 82 50 24 16 12 780 1 256 12 100 89 48 7 13 62 768 3 200 17 96 5 39 1277 12 2 600 3 192 8 80 2 32 57 8 12 384 1 156 1 78 4 24 8 4 4

Here are some S(2,5,65)s embedded as unitals in the known projective planes of order 16. Most of them are from Stoichev and Tonchev's paper.

### Designs S(2,6,66)

There are three non-isomorphic S(2,6,66) designs with automorphisms of order 13. Full automorphism groups are of order 39. The three designs can be distinguished by the number of complete quadrilaterals: 53053, 52884 and 53729 (complete quadrilaterals are sets of 4 lines intersecting in 6 points). These designs are not resolvable; it is not known whether there are any resolvable S(2,6,66)s.