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IEEE International Symposium on Information Theory (ISIT)

Some voltage graph-based LDPC tailbiting codes with large girth

Irina E. Bocharova, Florian Hug, Rolf Johannesson, Boris D. Kudryashov, Roman V. Satyukov

Published Jul 1, 2011• 8 citations
Publisher

Abstract

The relation between the parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and the biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Algorithms for searching iteratively for LDPC block codes with large girth are presented and constructions based on Steiner Triple Systems and short QC block codes are introduced, leading to new QC regular LDPC block codes with girth up to 24.

DOI
10.1109/isit.2011.6034230

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IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2011•4 citations
The relation between the parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and the biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Algorithms for searching iteratively for LDPC block codes with large girth are presented and constructions based on Steiner Triple Systems and short QC block codes are introduced, leading to new QC regular LDPC block codes with girth up to 24.
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The relation between low-density parity-check (LDPC) codes and hypergraphs supports searching for powerful LDPC codes based on hypergraphs. On the other hand, coding theory methods can be used in searching for hypergraphs with large girth. Moreover, compact representations of hypergraphs based on convolutional codes can be found. Algorithms for iteratively constructing LDPC codes with large girth and for determining their minimum distance are introduced. New quasi-cyclic (QC) LDPC codes are presented, some having both optimal girth and optimal minimum distance.
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Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
In this paper, some well-structured block designs are presented whose incidence matrices can be considered as the mother matrices of some regular and irregular column-weight three quasi cyclic low-density parity-check codes of girth 20. The class of LDPC codes derived from our construction possesses much more flexibility in terms of code length and rate. Moreover, the length of the regular constructed codes with rate at most 0.5 is at least 4.9 times smaller than the length of the voltage graph LDPC codes with the same rate and girth. Simulation results show that the constructed codes outperform progressive edge growth, repeat accumulate and random LDPC codes.
A new method using Hamming codes to construct base matrices of (J,K)-regular LDPC convolutional codes with large free distance is presented. By proper labeling the corresponding base matrices and tailbiting these parent convolutional codes to given lengths, a large set of quasi-cyclic (QC) (J,K)-regular LDPC block codes with large minimum distance is obtained. The corresponding Tanner graphs have girth up to 14. This new construction is compared with two previously known constructions of QC (J,K)-regular LDPC block codes with large minimum distance exceeding (J + 1)!. Applying all three constructions, new QC (J,K)-regular block LDPC codes with J = 3 or 4, shorter codeword lengths and/or better distance properties than those of previously known codes are presented.