A projective two-weight code related to the simple group ${\rm Co}_1$ of Conway

TL;DR

This paper constructs a binary projective two-weight code linked to the Conway group ${ m Co}_1$, explores its geometric and group-theoretic properties, and derives a new strongly regular graph from the codewords.

Contribution

It introduces a novel binary two-weight code associated with ${ m Co}_1$, analyzes its dual, and connects codewords to the Leech lattice and maximal subgroups.

Findings

The code is a faithful, absolutely irreducible submodule of a permutation module.

The dual code is a uniformly packed code with specific parameters.

A new strongly regular graph with 16,777,216 vertices is constructed.

Abstract

A binary $[98280, 24, 47104]_2$ projective two-weight code related to the sporadic simple group ${\rm Co}_1$ of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module induced by the primitive action of ${\rm Co}_1$ on the cosets of ${\rm Co}_2$. The dual code of this code is a uniformly packed $[98280, 98256,3]_2$ code. The geometric significance of the codewords of the code can be traced to the vectors in the Leech lattice, thus revealing that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${\rm Co}_1$. Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of ${\rm Co}_1$. As by-product, a new strongly regular graph on 16777216 vertices and valency 98280 is constructed using the codewords of the code.