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.