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

TL;DR

This paper extends the Mathieu group M23's permutation action to additive functions on a finite field of size 2^11, providing explicit matrix representations and tables for computational use.

Contribution

It introduces a novel explicit representation of M23 as additive functions on a finite field, including matrix forms for key generators and detailed tables.

Findings

Explicit 11x11 matrix representations for M23 generators.

Representation of M23 as additive functions on _{2^{11}}.

Provision of tables to facilitate future calculations.

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 $\varphi:\mathbb{F}_{2^{11}}\to \mathbb{F}_{2^{11}}$ such that $\varphi(x+y)=\varphi(x)+\varphi(y)$ and $\varphi|_{C}$ 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.