On the algebraic structure of quasi-cyclic codes I: Finite fields

Abstract

A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese Remainder Theorem (CRT), or of the Discrete Fourier Transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the recent (u + v|u - v) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.