Decomposing quasi-cyclic codes

Abstract

A new algebraic approach to quasi-cyclic codes is introduced. Technical tools include the Chinese Remainder Theorem, the Discrete Fourier Transform, Chain rings. The main results are a characterization of self-dual quasi-cyclic codes, a trace representation that generalizes that of cyclic codes, and an interpretation of the squaring and cubing construction (and of several similar combinatorial constructions). All 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.