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.

On idempotents of a class of commutative rings

“…is an idempotent element of the group ring Z m G, proving (12). Relation (13) also follows from Corollary 4.8.…”

p-Adic estimates of hamming weights in abelian codes over galois rings

“…The Fourier transform for Abelian codes over finite fields was introduced by MacWilliams [35] as a generalization of the Mattson-Solomon polynomial [36] for cyclic codes over finite fields. The basic theorems on the Fourier transform for Abelian codes over Galois rings are presented in [30], often without explicit proof, since the methods used for Abelian codes over finite fields and over generalize neatly.…”

Code–checkable group rings

“…When G is cyclic, this concept characterizes the classical cyclic codes over F as, in this case, the ideals of FG ∼ = F [x]/ < x n − 1 >. This concept has been first introduced by F. MacWilliams [7] in 1969. In general when G is abelian, they are called abelian codes.…”