$\mathrm{ EA}(q)$-additive Steiner 2-designs

TL;DR

This paper introduces new constructions for EA(q)-additive Steiner 2-designs, unifies existing methods, and presents novel designs, including resolvable and non-isomorphic examples, while also proving non-existence results for certain cyclic designs.

Contribution

The paper provides general construction methods for EA(q)-additive Steiner 2-designs, unifies known designs, and introduces new examples, expanding the understanding of additive combinatorial designs.

Findings

Constructed an EA(2^8)-additive 2-(52,4,1) design that is resolvable.

Found three non-isomorphic EA(3^5)-additive 2-(121,4,1) designs.

Proved that a cyclic 2-analog of a 2-(9,3,1) design cannot exist.

Abstract

A design is $G$-additive with $G$ an abelian group, if its points are in $G$ and each block is zero-sum in $G$. All the few known ``manageable" additive Steiner 2-designs are $\mathrm{EA}(q)$-additive for a suitable $q$, where $\mathrm{EA}(q)$ is the elementary abelian group of order $q$. We present some general constructions for $\mathrm{EA}(q)$-additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an additive $\mathrm{EA}(2^8)$-additive 2-$(52,4,1)$ design which is also resolvable, and three pairwise non-isomorphic $\mathrm{EA}(3^5)$-additive 2-$(121,4,1)$ designs, none of which is the point-line design of $\mathrm{PG}(4,3)$. In the attempt to find also an $\mathrm{EA}(2^9)$-additive 2-$(511,7,1)$ design, we prove that a putative 2-analog of a 2-$(9,3,1)$ design cannot be cyclic.