Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes

TL;DR

This paper introduces a new class of binary quasi-cyclic LDPC codes based on cyclic subgroups of finite fields, analyzing their rank properties and identifying subclasses with large minimum distances for efficient decoding.

Contribution

It presents a novel construction method for QC-LDPC codes using cyclic subgroups, along with rank analysis and identification of codes with large minimum distances.

Findings

Codes perform well over binary-input AWGN channel with SPA decoding

Provides combinatorial formulas for parity-check matrix ranks

Identifies subclasses with fast convergence and large minimum distances

Abstract

This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields. Experimental results show that the codes constructed perform well over the binary-input AWGN channel with iterative decoding using the sum-product algorithm (SPA). The second part analyzes the ranks of the parity-check matrices of codes constructed based on finite fields with characteristic of 2 and gives combinatorial expressions for these ranks. The third part identifies a subclass of constructed QC-LDPC codes that have large minimum distances. Decoding of codes in this subclass with the SPA converges very fast.