Quasi-cyclic LDPC codes based on pre-lifted protographs

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.

[1]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[2]  William S. Massey,et al.  Algebraic Topology: An Introduction , 1977 .

[3]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

[4]  Ieee Circuits,et al.  IEEE Transactions on Very Large Scale Integration (VLSI) Systems , 2018, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[5]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[6]  Sae-Young Chung,et al.  On the construction of some capacity-approaching coding schemes , 2000 .

[7]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[8]  D. Mackay,et al.  Evaluation of Gallager Codes for Short Block Length and High Rate Applications , 2001 .

[9]  Shu Lin,et al.  Low-density parity-check codes based on finite geometries: A rediscovery and new results , 2001, IEEE Trans. Inf. Theory.

[10]  Daniel A. Spielman,et al.  Efficient erasure correcting codes , 2001, IEEE Trans. Inf. Theory.

[11]  Daniel A. Spielman,et al.  Improved low-density parity-check codes using irregular graphs and belief propagation , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[12]  J. Thorpe.  Low-Density Parity-Check (LDPC) Codes Constructed from Protographs , 2003 .

[13]  Shu Lin,et al.  Near Shannon limit quasi-cyclic low-density parity-check codes , 2004, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[14]  T. Richardson,et al.  Multi-Edge Type LDPC Codes , 2004 .

[15]  Shu Lin,et al.  Near-Shannon-limit quasi-cyclic low-density parity-check codes , 2004, IEEE Trans. Commun..

[16]  Daniel J. Costello,et al.  LDPC block and convolutional codes based on circulant matrices , 2004, IEEE Transactions on Information Theory.

[17]  Marc P. C. Fossorier,et al.  Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices , 2004, IEEE Trans. Inf. Theory.

[18]  Lei Wei,et al.  Several properties of short LDPC codes , 2004, IEEE Trans. Commun..

[19]  Kyeongcheol Yang,et al.  Quasi-cyclic LDPC codes for fast encoding , 2005, IEEE Transactions on Information Theory.

[20]  Evangelos Eleftheriou,et al.  Regular and irregular progressive edge-growth tanner graphs , 2005, IEEE Transactions on Information Theory.

[21]  Shu Lin,et al.  Codes on finite geometries , 2005, IEEE Transactions on Information Theory.

[22]  Kyeongcheol Yang,et al.  Lifting methods for quasi-cyclic LDPC codes , 2006, IEEE Communications Letters.

[23]  Zongwang Li,et al.  Efficient encoding of quasi-cyclic low-density parity-check codes , 2006, IEEE Trans. Commun..

[24]  M. E. O'Sullivan,et al.  Algebraic construction of sparse matrices with large girth , 2006, IEEE Transactions on Information Theory.

[25]  Rüdiger L. Urbanke,et al.  Weight Distribution of Low-Density Parity-Check Codes , 2006, IEEE Transactions on Information Theory.

[26]  Sunghwan Kim,et al.  Quasi-Cyclic Low-Density Parity-Check Codes With Girth Larger Than $12$ , 2007, IEEE Transactions on Information Theory.

[27]  Zhongfeng Wang,et al.  Low-Complexity High-Speed Decoder Design for Quasi-Cyclic LDPC Codes , 2007, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[28]  Dariush Divsalar,et al.  Accumulate repeat accumulate codes , 2004, IEEE Global Telecommunications Conference, 2004. GLOBECOM '04..

[29]  Dariush Divsalar,et al.  Accumulate-Repeat-Accumulate Codes , 2007, IEEE Trans. Commun..

[30]  Shu Lin,et al.  Construction of Regular and Irregular LDPC Codes: Geometry Decomposition and Masking , 2007, IEEE Transactions on Information Theory.

[31]  Lajos Hanzo,et al.  Construction of Regular Quasi-Cyclic Protograph LDPC codes based on Vandermonde Matrices , 2008, 2008 IEEE 68th Vehicular Technology Conference.

[32]  L. Hanzo,et al.  Construction of Regular Quasi-Cyclic Protograph LDPC Codes Based on Vandermonde Matrices , 2008 .

[33]  Khaled A. S. Abdel-Ghaffar,et al.  A unified approach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fields , 2009, IEEE Transactions on Communications.

[34]  Ning Chen,et al.  Memory-efficient and high-throughput decoding of quasi-cyclic LDPC codes , 2009, IEEE Transactions on Communications.

[35]  Dariush Divsalar,et al.  Capacity-approaching protograph codes , 2009, IEEE Journal on Selected Areas in Communications.

[36]  Irina E. Bocharova,et al.  Graph-based convolutional and block LDPC codes , 2009, Probl. Inf. Transm..

[37]  Qin Huang,et al.  Quasi-cyclic LDPC codes: an algebraic construction , 2010, IEEE Transactions on Communications.

[38]  Qin Huang,et al.  Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares , 2010, IEEE Transactions on Communications.

[39]  Zhi Ding,et al.  Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields , 2011, IEEE Transactions on Communications.

[40]  Amir H. Banihashemi,et al.  Lowering the Error Floor of LDPC Codes Using Cyclic Liftings , 2011, IEEE Trans. Inf. Theory.

[41]  David G. M. Mitchell,et al.  Partially Quasi-Cyclic Protograph-Based LDPC Codes , 2011, 2011 IEEE International Conference on Communications (ICC).

[42]  Amir H. Banihashemi,et al.  Counting Short Cycles of Quasi Cyclic Protograph LDPC Codes , 2012, IEEE Communications Letters.

[43]  Roxana Smarandache,et al.  Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds , 2009, IEEE Transactions on Information Theory.

[44]  Florian Hug,et al.  Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth , 2011, IEEE Transactions on Information Theory.

[45]  Stark C. Draper,et al.  Hierarchical and High-Girth QC LDPC Codes , 2011, IEEE Transactions on Information Theory.

[46]  David G. M. Mitchell,et al.  Minimum Distance and Trapping Set Analysis of Protograph-Based LDPC Convolutional Codes , 2013, IEEE Transactions on Information Theory.

[47]  Jong-Seon No,et al.  Design of Multiple-Edge Protographs for QC LDPC Codes Avoiding Short Inevitable Cycles , 2012, IEEE Transactions on Information Theory.

[48]  Paul H. Siegel,et al.  Bounds on the Minimum Distance of Punctured Quasi-Cyclic LDPC Codes , 2012, IEEE Transactions on Information Theory.