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, the lifting degree, and the size and the structure of the base graph. As a result, for a given base graph, we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth. This also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree. The upper bounds derived here, which are applicable to both regular and irregular base graphs with no parallel edges, are in some cases more general and in some other cases tighter than the existing bounds. The results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth. As an example, we present new QC protograph LDPC code constructions with girth 8 using fully connected base graphs. These constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs.

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

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

[3]  Tor Helleseth,et al.  On the minimum distance of array codes as LDPC codes , 2003, IEEE Trans. Inf. Theory.

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

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

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

[7]  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.

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

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

[10]  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).

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

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

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

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

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

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

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

[18]  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.

[19]  Evangelos Eleftheriou,et al.  Rate-compatible low-density parity-check codes for digital subscriber lines , 2002, 2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577).

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

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

[22]  Amir H. Banihashemi,et al.  Design of Finite-Length Irregular Protograph Codes with Low Error Floors over the Binary-Input AWGN Channel Using Cyclic Liftings , 2012, IEEE Transactions on Communications.