Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4

Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge 4$ the associated collinearity graph contains maximal cliqu...

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Bibliographic Details
Main Authors Pankov, Mark, Petelczyc, Krzysztof, Żynel, Mariusz
Format Journal Article
LanguageEnglish
Published 28.06.2024
Subjects
Mathematics - Combinatorics
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Summary:Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge 4$ the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric $(n,2m,m)$-designs. The main results concern the case $k=4$ and give a geometric interpretation of the five well-known symmetric $(15,8,4)$-designs.
DOI:10.48550/arxiv.2406.19710