A combinatorial construction of an M_{12}-invariant code

TL;DR

This paper presents a new combinatorial construction of a ternary code with automorphism group M_{12}, utilizing the small Witt design, and confirms its minimum distance of 36, contributing to algebraic coding theory.

Contribution

It provides a novel combinatorial construction of an M_{12}-invariant code using the Steiner system S(5,6,12), linking group theory and design theory.

Findings

Constructed a ternary [66,10,36]_3-code with M_{12} symmetry.

Proved the code's minimum distance is 36.

Connected the code's structure to the small Witt design.

Abstract

In this work we summarized some recent results to be included in a forthcoming paper. 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. 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.