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arXiv · 2406.19710

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

Mark Pankov·Krzysztof Petelczyc·Mariusz Żynel
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Abstract

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.

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Mark Pankov, Krzysztof Petelczyc, Mariusz Żynel. 2024-06-28. Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$. https://arxiv.org/abs/2406.19710

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