The extended golay codes considered as ideals

Abstract

In this note we answer the following question in the affirmative: Is there a natural algebraic structure on the vector spaces containing the extended binary and ternary Golay codes such that the codes become ideals in these algebras? Our motivation was a note of J. Wolfmann, that describes the extended binary Golay code as the binary image of a principal ideal in a group algebra over the field with eight elements, and also a note of D. Y. Goldberg, that contains a related result for the extended ternary Golay code. In the following we construct those codes as ideals in the binary group algebra over the symmetric group H 4 and in the ternary twisted group algebra over the alternating group u 4, respectively. Before we present our results, we are going to remind the reader of the definition of Golay codes as special QR-codes. Here, for obvious reasons, we restrict out attention to the binary and ternary case. We assume that the reader is familiar with basic notions in coding theory and in representation theory as well.