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

TL;DR

This paper explores the geometric structure of symmetric (15,8,4)-designs through the lens of binary simplex codes of dimension 4, revealing new insights into their maximal cliques and geometric interpretation.

Contribution

It provides a geometric interpretation of symmetric (15,8,4)-designs using binary simplex codes and characterizes their maximal cliques within the associated collinearity graph.

Findings

Identifies maximal cliques corresponding to symmetric (15,8,4)-designs.

Provides a geometric interpretation of these designs.

Analyzes the case k=4 in detail.

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.