Spectral Analysis of Quasi-Cyclic Product Codes

Abstract

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row and column codes. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm. More specifically, we prove an explicit (unreduced) basis of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code in terms of the generator matrix in reduced Grobner basis with respect to the position-over-term (RGB/POT) order form of the $ {\ell _{A}}$ -quasi-cyclic row code and the $ {\ell _{B}}$ -quasi-cyclic column code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where $ {\ell _{A}}= {\ell _{B}}=1$ ). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code for two special cases: i) for $ {\ell _{A}}=2$ and $ {\ell _{B}}=1$ and ii) if the row code is a one-level $ {\ell _{A}}$ -quasi-cyclic code (for arbitrary $ {\ell _{A}}$ ) and $ {\ell _{B}}=1$ . For arbitrary $ {\ell _{A}}$ and $ {\ell _{B}}$ , the Pre-RGB/POT form of the generator matrix of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code is conjectured. The spectral analysis is applied to the generator matrix of the product of an $ {\ell }$ -quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given $ {\ell }$ -quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for $ {\ell }$ -phased burst errors with guaranteed decoding radius.