Partially Quasi-Cyclic Protograph-Based LDPC Codes

A significant amount of the analysis of protograph-based low-density parity-check (LDPC) codes has been devoted to the subclass of quasi-cyclic (QC) LDPC codes. Despite their implementation advantages and algebraic properties that make them easy to analyze, protograph-based QC-LDPC codes have undesirable fixed upper limits on important code parameters. This implies that picking a QC code from an asymptotically good or capacity approaching ensemble is suboptimal, since long QC codes will not perform close to the ensemble asymptotic limits. Indeed, these limits can only be achieved by codes that are not QC. In this paper we present an overview together with some new results on partially-QC protograph-based LDPC codes, i.e., LDPC codes whose parity-check matrix is partially composed of circulant submatrices. We perform both a minimum Hamming distance and girth analysis of these codes. Moreover, we present explicit partially-QC LDPC code constructions with parameters that exceed the restricted QC upper bounds.