Computation of Grobner Basis for Systematic Encoding of Generalized Quasi-Cyclic Codes

Vo TAM VAN  Hajime MATSUI  Seiichi MITA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E92-A   No.9   pp.2345-2359
Publication Date: 2009/09/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E92.A.2345
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Coding Theory
automorphism group,  Buchberger's algorithm,  division algorithm,  circulant matrix,  finite geometry low density parity check (LDPC) codes,  

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Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices; we call them echelon canonical form algorithm and transpose algorithm. Both algorithms require sufficiently small number of finite-field operations with the order of the third power of code-length. Each algorithm has its own characteristic. The first algorithm is composed of elementary methods and is appropriate for low-rate codes. The second algorithm is based on a novel formula and has smaller computational complexity than the first one for high-rate codes with the number of orbits (cyclic parts) less than half of the code length. Moreover, we show that a serial-in serial-out encoder architecture for FG LDPC codes is composed of linear feedback shift registers with the size of the linear order of code-length; to encode a binary codeword of length n, it takes less than 2n adder and 2n memory elements.