Counting Short Cycles of Quasi Cyclic Protograph LDPC Codes

An efficient method for counting short cycles in the Tanner graphs of quasi cyclic (QC) protograph low-density parity-check (LDPC) codes is presented. The method is based on the relationship between the number of short cycles in the graph and the eigenvalues of the directed edge matrix of the graph. We demonstrate that for a QC protograph LDPC code, the complexity of computing such eigenvalues can be reduced significantly by representing the directed edge matrix as a block circulant matrix. Numerical results are presented to show the lower complexity of the proposed method compared to the existing algorithms for counting short cycles. These results also reveal that QC LDPC codes on average have a superior short cycle and girth distribution compared to similar randomly constructed codes.

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

[2]  A. Terras,et al.  Zeta functions of finite graphs and coverings, III , 1996 .

[3]  Keith M. Chugg,et al.  An algorithm for counting short cycles in bipartite graphs , 2006, IEEE Transactions on Information Theory.

[4]  D. West.  Introduction to Graph Theory , 1995 .

[5]  A. Terras,et al.  Zeta Functions of Finite Graphs and Coverings , 1996 .

[6]  Amir H. Banihashemi,et al.  A heuristic search for good low-density parity-check codes at short block lengths , 2001, ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No.01CH37240).

[7]  Hua Xiao,et al.  Error rate estimation of low-density parity-check codes on binary symmetric channels using cycle enumeration , 2009, IEEE Transactions on Communications.

[8]  Amir H. Banihashemi,et al.  A Message-Passing Algorithm for Counting Short Cycles in a Graph , 2010, ArXiv.

[9]  Jun Fan,et al.  A Method of Counting the Number of Cycles in LDPC Codes , 2006, 2006 8th international Conference on Signal Processing.

[10]  Noga Alon,et al.  Finding and counting given length cycles , 1997, Algorithmica.

[11]  Guozhen Xiao,et al.  Relation Between Parity-Check Matrixes and Cycles of Associated Tanner Graphs , 2007, IEEE Communications Letters.

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

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

[14]  G. Tee.  Eigenvectors of block circulant and alternating circulant matrices , 2005 .

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

[16]  Raf Vandebril,et al.  Matrix Computations and Semiseparable Matrices: Eigenvalue and Singular Value Methods , 2008 .

[17]  Jörg Flum,et al.  The Parameterized Complexity of Counting Problems , 2004, SIAM J. Comput..

[18]  R. M. Tanner,et al.  A Class of Group-Structured LDPC Codes , 2001 .