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IEEE International Symposium on Information Theory (ISIT)

A greedy search for improved QC LDPC codes with good girth profile and degree distribution

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

Published Jul 1, 2012• 5 citations
Publisher

Abstract

The girth profile is introduced and search algorithms for regular and irregular quasi-cyclic LDPC block codes with both good girth profile and good degree distribution are presented. New QC LDPC block codes of various code rates are obtained and their bit error rate performance is compared with that of the corresponding LDPC block codes defined in the IEEE 802.16 WiMAX standard of the same block length and code rate.

DOI
10.1109/isit.2012.6284129

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The girth profile is introduced and search algorithms for regular and irregular quasi-cyclic LDPC block codes with both good girth profile and good degree distribution are presented. New QC LDPC block codes of various code rates are obtained and their bit error rate performance is compared with that of the corresponding LDPC block codes defined in the IEEE 802.16 WiMAX standard of the same block length and code rate.
Techniques for searching for good quasi-cyclic (QC) LDPC block codes of short and moderate lengths which are suitable for practical purposes are studied. To facilitate implementations only codes whose parity-check matrices having bidiagonal structure of their submatrices and consequently having low encoding complexity are considered. The problem of finding QC LDPC codes with the near-optimum frame or bit error rate performance is split into two independent steps: searching for the near-optimum column degree distribution of the parity-check matrix together with the best base matrix for this degree distribution and searching for the near-optimum labeling of the chosen base matrix. Sets of parameters and criteria for both steps are introduced and discussed. They allow further reduction of the search complexity without significant loss of the search optimality. New QC LDPC block codes of various code rates are obtained and their BER and FER performances are compared with those of the LDPC block codes as well as the turbo codes defined in the IEEE 802.16 WiMAX standard.
Irregular QC LDPC codes with parity-check matrices having different degree distributions are studied. A new algorithm for finding regular and irregular QC LDPC codes with a good girth profile as well as a good sliding-window girth is presented. As examples, simulation results for QC LDPC codes with good girth profile, rate R=4/5, and lengths about 1000, 2000, and 4000, constructed from base matrices with proper degree distributions are given. Their simulated BER and FER performances for belief propagation decoding are compared with the best previously known irregular QC LDPC codes of the same rate and length. It is shown that the constructed codes outperform the best previously known codes of same rate and lengths.
The QC(quasi-cyclic) LDPC codes are an important subclass of LDPC codes. The key to improving the performance of QC LDPC codes is how to construct a parity-check matrix H with a girth distribution as good as possible. In this paper, method of girth detection of QC LDPC codes is described first. Furthermore, a novel algorithm, GirthOpt-PSO algorithm, is proposed to construct the QC LDPC codes, which achieves a good girth distribution based on the PSO algorithm. Simulation results show that the performance of the QC LDPC codes constructed by proposed algorithm is better than Tanner code and Array codes. Besides, the proposed algorithm is more flexible for designing the QC LDPC codes with desired block length and rate as well as good girth.
In this paper, we study the cycle structure of quasi-cyclic (QC) low-density parity-check (LDPC) codes with the goal of obtaining the shortest code with a given degree distribution and girth. We focus on QC-LDPC codes, whose Tanner graphs are cyclic liftings of fully connected base graphs of size 3 × n, n ≥ 4, and obtain minimal lifting degrees that result in girths 6 and 8. This is performed through an efficient exhaustive search, and as a result, we also find all the possible non-isomorphic codes with the same minimum block length, girth, and degree distribution. The exhaustive search, which is ordinarily a formidable task, is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process. Many of the pruning techniques proposed in this paper are also applicable to QC-LDPC codes with base graphs other than the 3 × n fully connected ones discussed here, as well as to codes with a larger girth. To further demonstrate the effectiveness of the pruning techniques, we use them to search for QC-LDPC codes with girths 10 and 12, and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature. In addition, motivated by the exhaustive search results, we tighten the lower bound on the block length of QC-LDPC codes of girth 6 constructed from fully connected 3 × n base graphs, and construct codes that achieve the lower bound for an arbitrary value of n ≥ 4.
In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.
We search for good regular quasi-cyclic (QC) LDPC codes with J = 2 ones in each column. In order to simplify the search, QC LDPC codes are represented in the form of tail-biting (TB) convolutional codes. A modified BEAST algorithm is used for finding the free distance (minimum distance) and the girth of both parent convolutional and block LDPC codes. Representations of known bipartite graphs and LDPC based on finite geometries in the form of TB convolutional codes are found. This approach is further generalized for J = 3 QC LDPC codes. Examples of good short LDPC codes with large girth and minimum distance are given. For example, we present a rate 2=5 J = 3 QC LDPC (225, 92)- code with girth 8 and minimum distance 24.