On the girth of quasi cyclic protograph LDPC codes

In this paper, we study the relationships between the girth of the Tanner graph of a quasi cyclic (QC) protograph low-density parity-check (LDPC) code, on one hand, and the lifting degree and the size and the structure of the base graph, on the other hand. As a result, for a given base graph and a given lifting degree, we derive an upper bound on the girth of the resulting lifted graphs (codes). The upper bounds derived here are generally tighter than the existing bounds. The results presented in this work can be used to select an appropriate lifting degree for a given base graph, in order to have a desired girth, or to provide some insight in designing good base graphs, or to properly select the base graph's edge permutations.

[1]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[2]  C. Kelley,et al.  On codes designed via algebraic lifts of graphs , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[3]  Bane Vasic,et al.  Coding and Signal Processing for Magnetic Recording Systems , 2004 .

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

[5]  Sunghwan Kim,et al.  On the girth of tanner (3, 5) quasi-cyclic LDPC codes , 2006, IEEE Transactions on Information Theory.

[6]  Christine A. Kelley,et al.  LDPC codes from voltage graphs , 2008, 2008 IEEE International Symposium on Information Theory.

[7]  Xinmei Wang,et al.  Girth-12 Quasi-Cyclic LDPC Codes with Consecutive Lengths , 2010, ArXiv.

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

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

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

[11]  Florian Hug,et al.  Some voltage graph-based LDPC tailbiting codes with large girth , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

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

[13]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[14]  Amir H. Banihashemi,et al.  Message-Passing Algorithms for Counting Short Cycles in a Graph , 2010, IEEE Transactions on Communications.

[15]  J. Yedidia,et al.  Construction of high-girth QC-LDPC codes , 2008, 2008 5th International Symposium on Turbo Codes and Related Topics.

[16]  Amir H. Banihashemi,et al.  An efficient algorithm for finding dominant trapping sets of LDPC codes , 2011, 2010 6th International Symposium on Turbo Codes & Iterative Information Processing.

[17]  Amir H. Banihashemi,et al.  Design of irregular quasi-cyclic protograph codes with low error floors , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

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

[19]  Navin Kashyap,et al.  Shortened Array Codes of Large Girth , 2005, IEEE Transactions on Information Theory.

[20]  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).

[21]  M. Esmaeili,et al.  Structured quasi-cyclic LDPC codes with girth 18 and column-weight J⩾3 , 2010 .

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

[23]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

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

[25]  Florian Hug,et al.  Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth , 2011, 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]  Tor Helleseth,et al.  On the minimum distance of array codes as LDPC codes , 2003, IEEE Trans. Inf. Theory.