hrvatski

Super-regular Steiner 2-designs

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 2^n3 ≥12. The case k≡2(mod4) is a genuine exception whereas k=2^n3 ≥12 is at the moment a possible exception. We also find super-regular 2-(p^n, p, 1)designs with p ∈{;5, 7}; and n ≥3 which are not isomorphic to the point-line design of AG(n, p).

(Strictly) additive design ; Steiner 2-design ; Automorphism group ; Regular design ; (Strong) difference family ; Cyclotomy ; Difference matrix

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