Binary Codes Which Are Ideals in the Group Algebra of an Abelian Group

Abstract

A cyclic code is an ideal in the group algebra of a special kind of Abelian group, namely a cyclic group. Many properties of cyclic codes are special cases of properties of ideals in an Abelian group algebra. A character of an Abelian group G of order v is, for our purposes, a homomorphism of G into the group of vth roots of unity over GF(2). If G is cyclic with generator x, the character is entirely determined by what it does to x; this effect is kept, and the characters are discarded. If G is not cyclic it is necessary to rehabilitate the characters. Without them the notation is impossible; with them one can prove a number of theorems which reduce in the special case to well-known properties of cyclic codes. Moreover the writer thinks that the general proof is often easier and more suggestive than the proof for the special case. To support this point of view we produce a new theorem, which of course also applies to cyclic codes.