Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting. However, these restrictions impose undesirable fixed upper limits on important code parameters, such as minimum distance and girth. In this paper, we consider an approach to constructing QC-LDPC codes that uses a two-step lifting procedure based on a protograph, and, by following this method instead of the usual one-step procedure, we obtain improved minimum distance and girth properties. We also present two new design rules for constructing good QC-LDPC codes using this two-step lifting procedure, and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared to one-step circulant-based liftings. The expected performance improvement is verified by simulation results.Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting. However, these restrictions impose undesirable fixed upper limits on important code parameters, such as minimum distance and girth. In this paper, we consider an approach to constructing QC-LDPC codes that uses a two-step lifting procedure based on a protograph, and, by following this method instead of the usual one-step procedure, we obtain improved minimum distance and girth properties. We also present two new design rules for constructing good QC-LDPC codes using this two-step lifting procedure, and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared to one-step circulant-based liftings. The expected performance improvement is verified by simulation results.

arXiv:1407.5364Jul 21, 20141 citation

Constructing good QC-LDPC codes by pre-lifting protographs

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

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are of great interest to code designers because of their implementation advantages and algebraic properties that facilitate their analysis. In this paper, we present some new results on QC-LDPC codes that are constructed using a two-step lifting procedure based on a protograph, and, by implementing this method instead of the usual one-step procedure, we are able to show improved minimum distance and girth properties. We also present two design rules to construct QC-LDPC codes: one uses only circulant permutation matrices at the first (pre-lifting) stage and the other uses a selection of non-commuting permutation matrices. For both techniques, we obtain a demonstrable increase in the minimum distance compared to a one-step circulant-based lifting. The expected performance improvement is verified by simulation results.Quasi-cyclic (QC) low-density parity-check (LDPC) codes are of great interest to code designers because of their implementation advantages and algebraic properties that facilitate their analysis. In this paper, we present some new results on QC-LDPC codes that are constructed using a two-step lifting procedure based on a protograph, and, by implementing this method instead of the usual one-step procedure, we are able to show improved minimum distance and girth properties. We also present two design rules to construct QC-LDPC codes: one uses only circulant permutation matrices at the first (pre-lifting) stage and the other uses a selection of non-commuting permutation matrices. For both techniques, we obtain a demonstrable increase in the minimum distance compared to a one-step circulant-based lifting. The expected performance improvement is verified by simulation results.

IEEE Information Theory Workshop (ITW)Sep 1, 20124 citations

Quasi-cyclic LDPC codes based on pre-lifted protographs

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

Quasi-cyclic Low-Density Parity-Check (QC-LDPC) codes based on protographs are of great interest to code designers because of their implementation advantages and algebraic properties that make them easy to analyze. However, the protograph structure imposes undesirable fixed upper limits on important code parameters. In this paper, we show that the upper bound on the minimum Hamming distance of protograph-based QC codes can be improved by the careful application of a two-step lifting procedure applied to the protograph. The promised improvement is validated by constructing codes with minimum distance exceeding the upper bound for QC codes based on a particular protograph.Quasi-cyclic Low-Density Parity-Check (QC-LDPC) codes based on protographs are of great interest to code designers because of their implementation advantages and algebraic properties that make them easy to analyze. However, the protograph structure imposes undesirable fixed upper limits on important code parameters. In this paper, we show that the upper bound on the minimum Hamming distance of protograph-based QC codes can be improved by the careful application of a two-step lifting procedure applied to the protograph. The promised improvement is validated by constructing codes with minimum distance exceeding the upper bound for QC codes based on a particular protograph.

IEEE Information Theory Workshop (ITW)Oct 1, 20119 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

Design of Multiple-Edge Protographs for QC LDPC Codes Avoiding Short Inevitable Cycles

Hosung Park, Seokbeom Hong, Jong-Seon No +1 more

There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them focus on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple (i.e., parallel) edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on a graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and another method is proposed for regular QC LDPC codes with girth at least 14. Moreover, a construction algorithm of QC LDPC codes based on certain liftings of multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes with progressive edge-growth (PEG) LDPC codes and with PEG QC LDPC codes.There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them focus on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple (i.e., parallel) edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on a graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and another method is proposed for regular QC LDPC codes with girth at least 14. Moreover, a construction algorithm of QC LDPC codes based on certain liftings of multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes with progressive edge-growth (PEG) LDPC codes and with PEG QC LDPC codes.

arXiv:1210.3906IEEE Transactions on Information TheoryJun 14, 201330 citations

On the Construction of Multitype Quasi-Cyclic Low-Density Parity-Check Codes With Different Girth and Length

Farzaneh Abedi, Mohammad Gholami

Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girth $g_{\max }$ of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth $g_{\max }$ , some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth $g$ , $g\le g_{\max }$ , have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girth$g_{\max }$of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth$g_{\max }$, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth$g$,$g\le g_{\max }$, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.

IEEE AccessJan 1, 20212 citations

g_{\max }

of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth

g_{\max }

, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth

g

g\le g_{\max }

, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.