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 a paper in Glasnik Matematicki 37(57) (2002), p. 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 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 a paper under preparation automorphisms of order 2 and 3 are studied and used to find many more examples. Here is a list of some 50000 non-isomorphic S(2,4,37) designs (51402 to be precise). 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), p. 199-212). Below is the distribution by full automorphism group order.

  |Aut|     Freq.  
1111
544
371
334
272
187
11280
9203
31748
249152

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), p. 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(2,6,66)

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.

Further S(2,6,66) designs could have automorphisms of order 2, 3, 5 and 11. I am currently looking for more examples with automorphisms of order 11.

Vedran Krcadinac, 17.1.2004.