Dual-Diagonal QC-LDPC Codes Free of Small Cycles From GCD Framework via Negative Coefficients

A novel class of dual-diagonal (DD) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight is proposed. For the exponent matrix of such a DD-QC-LDPC code, the left sub-matrix is explicitly generated via new formulae which are motivated by a search algorithm based on the greatest-common-divisor (GCD) framework and a new strategy allowing negative coefficients. The right sub-matrix (DD matrix) is designed by modifying the existing DD matrix while preserving the simple encoding property directly through the parity-check matrix. Performance simulations indicate that the proposed DD-QC-LDPC codes significantly outperform several existing counterparts with or without DD structure. Unlike existing formulae which involve only non-negative coefficients, the new formulae motivated by proposed search algorithm allow for negative coefficients, which lead to significantly improved lower bounds on circulant sizes for column weights from five to seven. As a result, the new DD-QC-LDPC codes not only enable simple encoding and low description complexity, but also possess consecutive and reasonably small circulant sizes, flexible sizes for exponent matrices as well as good decoding performance.