Quasi-Cyclic LDPC Codes With Parity-Check Matrices of Column Weight Two or More for Correcting Phased Bursts of Erasures

In his pioneering work on LDPC codes, Gallager dismissed codes with parity-check matrices of weight two after proving that their minimum Hamming distances grow at most logarithmically with their code lengths. In spite of their poor minimum Hamming distances, it is shown that quasi-cyclic LDPC codes with parity-check matrices of column weight two have good capability to correct phased bursts of erasures which may not be surpassed by using quasi-cyclic LDPC codes with parity-check matrices of column weight three or more. By modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. Quasi-cyclic LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel are also considered. Examples of such codes based on Reed-Solomon and Gabidulin codes are presented.

I. Introduction

With the rediscovery of low-density parity-check (LDPC) codes by the turn of the century, researchers have recognized that LDPC codes perform well over the binary erasure channel (BEC) that causes the value of a transmitted bit to be lost, in addition to their superior performance over the AWGN channel [1]. A simple “peeling” algorithm that can be applied to a sparse parity-check matrix of an LDPC code to correct erasures was proposed early on. The algorithm may not correct all erasures that can be corrected by an optimal maximum-likelihood (ML) decoder. However, for long LDPC codes, it is very difficult to determine the capability of an ML decoder to correct erasures let alone implement such a decoder. Motivated by potential applications of LDPC codes in storage systems and communication over fading channels, researchers investigated the capability of LDPC codes to correct erasure bursts, and in particular one long burst of erasures [2]–[6].