Analysis of quasi-cyclic low-density parity-check codes and protograph codes

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices (or circulants). In this correspondence, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i, i = 6, 7, 8, 9,10, i.e., the cycles that always exist regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g, g > 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14 or 18. We also propose a couple of shift value assigning rules for circulants of a QC LDPC code guaranteeing the girth 14.A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices (or circulants). In this correspondence, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i, i = 6, 7, 8, 9,10, i.e., the cycles that always exist regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g, g > 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14 or 18. We also propose a couple of shift value assigning rules for circulants of a QC LDPC code guaranteeing the girth 14.

IEEE Transactions on Information TheoryAug 1, 200795 citations

Quantum quasi-cyclic low-density parity-check codes

Dazu Huang, Zhigang Chen, Xin Li +1 more

Sep 16, 20095 citations

Partially Quasi-Cyclic Protograph-Based LDPC Codes

Roxana Smarandache, David G. M. Mitchell, Danniel J. Costello

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.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.

International Conference on CommunicationsJun 1, 201110 citations

Cycle Analysis in Expanded QC LDPC Codes

Xiao Rong Gao, Nan Zhang

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices. In this paper, we use permutation matrices to form protograph LDPC codes. In protograph LDPC codes, we present cycle relationships of the mother matrix and the protograph LDPC code. It is derived that the girth of a protograph LDPC code is no smaller than its mother matrix .And cycles in the protograph LDPC codes are uniquely induced by the cycles in its mother matrix.A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices. In this paper, we use permutation matrices to form protograph LDPC codes. In protograph LDPC codes, we present cycle relationships of the mother matrix and the protograph LDPC code. It is derived that the girth of a protograph LDPC code is no smaller than its mother matrix .And cycles in the protograph LDPC codes are uniquely induced by the cycles in its mother matrix.

Jan 1, 20110 citations

PERFORMANCE OF PSEUDO-RANDOM AND QUASI-CYCLIC LOW DENSITY PARITY CHECK CODES

Onur Hüsnü Kazanci

Jan 1, 20071 citation