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arXiv · 2511.01073

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

Marco Buratti·Mario Galici·Alessandro Montinaro·Anamari Nakic·Alfred Wassermann
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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.

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Marco Buratti, Mario Galici, Alessandro Montinaro, Anamari Nakic, Alfred Wassermann. 2025-11-02. $\mathrm{ EA}(q)$-additive Steiner 2-designs. https://arxiv.org/abs/2511.01073

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