Abelian group factorization from cyclic and quasi-cyclic codes

Abstract

In this paper, we use the algebraic structures of cyclic codes and algorithmic techniques to find factorizations of abelian groups from cyclic codes. We construct specific subclasses of quasi-cyclic codes and provide the conditions with which we obtain a normalized factorization of certain abelian groups. The factorization, in both cases, is constituted by two sets, one corresponding to the cyclic code and the other corresponding to the words that represent all possible error polynomials of the cyclic code besides the zero vector.