Algebraic Constructions of Girth-Eight Quasi-Cyclic Matrices with a Multitude of Redundant Rows

Abstract

The absence of small cycles in the Tanner graph and the presence of redundant rows in the parity-check matrix (PCM) are two main tactics to improve the decoding performance of a quasi-cyclic (QC) low-density parity-check (LDPC) code from the construction point of view. Existing systematic methods to yield a PCM with a multitude of redundant rows impose very severe restrictions on the circulant size and guarantee the elimination of only 4-cycles. By exploiting the greatest-common-divisor (GCD) framework, several existing methods are identified and two new methods are proposed to generate PCMs with relatively flexible circulant sizes, a considerable number of redundant rows and no 4-cycles and 6-cycles in the associated Tanner graphs. Moreover, the rank of the PCM associated with each of these methods is theoretically proved for odd circulant sizes. Simulation results show that, the proposed QC-LDPC codes can achieve better decoding performance compared with those from the progressive-edge-growth (PEG) and search-based symmetrical methods whose PCMs contain basically no redundant rows.