Hierarchical and High-Girth QC LDPC Codes

• Published in: IEEE transactions on information theory Vol. 59; no. 7; pp. 4553 - 4583
• Main Authors: Yige Wang, Draper, S. C., Yedidia, J. S.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Charge carrier processes; Codes; Coding theory; Coding, codes; Decoding; Educational institutions; Error correction codes; Exact sciences and technology; Information theory; Information, signal and communications theory; Low density parity check codes; low-density parity-check (LDPC) codes; Parity check codes; Polynomials; protograph; quasi-cyclic codes; Signal and communications theory; Signal to noise ratio; Telecommunications and information theory; Cyclic code; High density; Mapping; Algorithm; quasi-cyclic codes; Implementation; Hierarchized structure; low-density parity-check (LDPC) codes; Parity check codes; Error correcting code; Error correction codes; protograph; Parity check
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2253512

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.