**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 Gröbner 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.