The Pace code, the Mathieu group $M_{12}$ and the small Witt design $S(5,6,12)$

Abstract

A ternary ${[66,10,36]}_3$-code admitting the Mathieu group $M_{12}$ as a group of automorphisms has recently been constructed by N. Pace, see Pace (2014). We give a construction of the Pace code in terms of $M_{12}$ as well as a combinatorial description in terms of the small Witt design, the Steiner system $S(5,6,12)$. We also present a proof that the Pace code does indeed have minimum distance $36.$