A Self-Orthogonal Doubly-Even Code Invariant Under \(M^cL\)

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 \(M^c L\).