The Mathieu group $M_{23}$ as additive functions on the finite field of size ${2^{11}}$

Abstract

We explicitly extend the standard permutation action of the Mathieu group $M_{23}$ on a 23 element set $C=C_{23}$ contained in a finite field of $2^{11}$ elements $\mathbb{F}_{2^{11}}$ to additive functions on this finite field. That is we represent $M_{23}$ as functions $φ:\mathbb{F}_{2^{11}}\to \mathbb{F}_{2^{11}}$ such that $φ(x+y)=φ(x)+φ(y)$ and $φ|_{C}$ is the standard permutation action. We give explicit $11\times 11$ matrices for the pair of standard generators of order $23$ and order $5$, as well as many tables to help facilitate future calculations.