  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.
