$\mathrm{ EA}(q)$-additive Steiner 2-designs
Marco Buratti, Mario Galici, Alessandro Montinaro, Anamari Nakic, Alfred Wassermann
TL;DR
The paper investigates EA($q$)-additive Steiner $2$-designs, introducing general constraints on admissible $q$, and develops constructive approaches yielding cyclic or $1$-rotational additive designs. It delivers concrete new examples: an EA($2^8$)-additive $2$-$(52,4,1)$ design that is $1$-rotational and resolvable, and at least four pairwise non-isomorphic EA($3^5$)-additive $2$-$(121,4,1)$ designs, including one corresponding to PG$(4,3)$. It also tackles the existence of an additive $(511,7,1)$ design, showing that a cyclic $2$-analog of $(9,3,1)$ cannot exist via Kramer–Mesner computations and a complementary geometric argument, supported by a detailed algorithmic search. The work highlights both new constructive instances and limitations of additive designs, underscoring the role of computational methods in exploring q-analogs and related combinatorial structures.
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.