Super-regular Steiner 2-designs

Abstract

A design is additive under an abelian group G (briefly, G-additive) if, up to isomorphism, its point set is contained in G and the elements of each block sum up to zero. The only known Steiner 2-designs that are G-additive for some G have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces AG(n,q), the point-line designs of the projective planes PG(2,q), the point-line designs of the projective spaces PG(n,2) and a sporadic example of a 2-(8191,7,1) design. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly G-additive (the point set is exactly G) and G-regular (any translate of any block is a block as well) at the same time. These designs will be called “G-super-regular”. Our main result is that there are infinitely many values of v for which there exists a super-regular, and therefore additive, 2-(v,k,1) design whenever k is neither singly even nor of the form 2n3≥12. The case k≡2 (mod 4) is a genuine exception whereas k=2n3≥12 is at the moment a possible exception. We also find super-regular 2-(pn,p,1) designs with p∈{5,7} and n≥3 which are not isomorphic to the point-line design of AG(n,p).