Super-regular Steiner 2-designs

TL;DR

This paper introduces the concept of super-regular Steiner 2-designs, establishing their existence for infinitely many parameters and identifying cases where such designs differ from known affine and projective space designs.

Contribution

The paper defines super-regular Steiner 2-designs and proves their existence for infinitely many parameters, expanding the class of known additive designs beyond classical affine and projective space configurations.

Findings

Existence of super-regular 2-(v,k,1) designs for infinitely many v when k is neither singly even nor of the form 2^n3≥12.

Identification of specific super-regular designs with p in {5,7} and n≥3 that are not isomorphic to affine space point-line designs.

Clarification of exceptions for certain block sizes, notably k≡2 (mod 4).

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 $PG(2,q)$, and the point-line designs of the projective spaces $PG(n,2)$. 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\break "$G$-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 $2^n3\geq12$. The case $k\equiv2$ (mod 4) is a definite exception whereas $k=2^n3\geq12$ is at the moment a possible exception. We also find super-regular $2$-$(p^n,p,1)$ designs with $p\in\{5,7\}$ and $n\geq3$ which are not isomorphic to the point-line design of $AG(n,p)$.