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 the column-code. 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 l\a l\b-quasi-cyclic product code in terms of the generator matrix in reduced Gr\"obner basis with respect to the position-over-term order (RGB/POT) form of the l\a-quasi-cyclic row- and the l\b-quasi-cyclic column-code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where l\a =l\b=1). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an l\a l\b-quasi-cyclic product code for two special cases: (i) for l\a=2 and l\b=1, and (ii) if the row-code is a 1-level l\a-quasi-cyclic code (for arbitrary l\a) and l\b=1.For arbitrary l\a and l\b, the Pre-RGB/POT form of the generator matrix of an l\a l\b-quasi-cyclic product code is conjectured.The spectral analysis is applied to the generator matrix of the product of an l-quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given l-quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for l-phased burst errors with guaranteed decoding radius.
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