A Self-Orthogonal Doubly-Even Code Invariant Under $M^{c}L$

Abstract

We examine a design $\mathcal{D}$ and a binary code $C$ constructed from a primitive permutation representation of degree $2025$ of the sporadic simple group $M^{c}L$. We prove that $\text{Aut}(C)=\text{Aut}(\mathcal{D})=M^{c}L$ and determine the weight distribution of the code and that of its dual. In Section $6$ we show that for a word $w_i$ of weight $7$, where $i\in \{848,896,912,972,1068,1100,1232,1296\}$ the stabilizer $(M^{\circ }L{)}_{w_i}$ is a maximal subgroup of $M^{\circ }L$. The words of weight $1024$ split into two orbits $C_{(1024{)}_1}$ and $C_{(1024{)}_2}$, respectively. For $w_i\in C_{(1024{)}_1}$, we prove that $(M^{c}L{)}_{w_i}$ is a maximal subgroup of $M^{c}L$.