Hierarchical and High-Girth QC LDPC Codes

We present an approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation, and compatible with some desired input code structure defined using a protograph. The approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi-cyclic (HQC) LDPC codes. Whereas the parity check matrices of standard quasi-cyclic (QC) LDPC codes are composed of circulant submatrices, those of HQC LDPC codes are composed of a hierarchy of circulant submatrices that are, in turn, constructed from circulant submatrices, and so on, through some number of levels. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code, subject to bounds imposed by the fact that HQC codes are still quasi-cyclic. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a “squashing” procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with three design examples of QC LDPC codes—two girth-10 codes of rates $1/3$ and 0.45 and one girth-8 code of rate 0.7—all of which are obtained from protographs of one-sided spatially coupled codes.