Spectral Analysis of Quasi-Cyclic Product Codes

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 ℓ
_A
ℓ
_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 ℓ
_A
-quasi-cyclic row code and the ℓ
_B
-quasicyclic 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 one-level ℓ
_A
-quasi-cyclic code (for arbitrary ℓ
_A
) and ℓ
_B
= 1. For arbitrary ℓ
_A
and ℓ
_B
, the PreRGB/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.