Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes

TL;DR

This paper provides a comprehensive analysis of the rank and row-redundancy of circulant matrix arrays used in QC-LDPC codes, introducing tight bounds, exact rank expressions, and new constructions to enhance code performance.

Contribution

It introduces tight bounds on rank and row-redundancy, derives exact rank formulas for specific constructions, and proposes new QC-LDPC code designs with improved redundancy properties.

Findings

Derived tight upper bounds on rank and row-redundancy.

Provided combinatorial expressions for exact rank in specific cases.

Presented new QC-LDPC code constructions with large row-redundancy.

Abstract

This paper is concerned with general analysis on the rank and row-redundancy of an array of circulants whose null space defines a QC-LDPC code. Based on the Fourier transform and the properties of conjugacy classes and Hadamard products of matrices, we derive tight upper bounds on rank and row-redundancy for general array of circulants, which make it possible to consider row-redundancy in constructions of QC-LDPC codes to achieve better performance. We further investigate the rank of two types of construction of QC-LDPC codes: constructions based on Vandermonde Matrices and Latin Squares and give combinatorial expression of the exact rank in some specific cases, which demonstrates the tightness of the bound we derive. Moreover, several types of new construction of QC-LDPC codes with large row-redundancy are presented and analyzed.