Computation of Grobner basis for systematic encoding of generalized quasi-cyclic codes

TL;DR

This paper introduces two algorithms for computing Grobner bases of GQC codes from their parity check matrices, enabling systematic encoding and efficient implementation for various code types.

Contribution

It proposes novel algorithms for Grobner basis computation of GQC codes, improving encoding efficiency and applicability across different code classes.

Findings

Algorithms require polynomially small finite-field operations

The first algorithm uses elementary methods, the second is faster for high-rate codes

Encoder architecture for FG LDPC codes uses linear feedback shift registers

Abstract

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: 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 the second algorithm is based on a novel formula and is faster than the first one for high-rate codes. 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. Keywords: automorphism group, Buchberger's algorithm, division algorithm, circulant matrix, finite geometry low density parity check (LDPC) codes.