Quasi-Cyclic Asymptotically Regular LDPC Codes

TL;DR

This paper explores the properties of quasi-cyclic LDPC codes derived from asymptotically regular ensembles, demonstrating how to improve their minimum Hamming distance bounds through careful protograph design and circulant arrays.

Contribution

It introduces methods to enhance the minimum distance bounds of QC LDPC codes by selecting component protographs and using circulant arrays in graph covers.

Findings

Improved upper bounds on minimum Hamming distance for QC LDPC codes.

Careful protograph selection enhances code properties.

Circulant arrays in graph covers further improve minimum distance bounds.

Abstract

Families of "asymptotically regular" LDPC block code ensembles can be formed by terminating (J,K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates, minimum distance that grows linearly with block length, and capacity approaching iterative decoding thresholds, despite the fact that the terminated ensembles are almost regular. In this paper, we investigate the properties of the quasi-cyclic (QC) members of such an ensemble. We show that an upper bound on the minimum Hamming distance of members of the QC sub-ensemble can be improved by careful choice of the component protographs used in the code construction. Further, we show that the upper bound on the minimum distance can be improved by using arrays of circulants in a graph cover of the protograph.