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International Symposium on Turbo Codes and Iterative Information Processing

(3, L) quasi-cyclic LDPC codes: Simplified exhaustive search and designs

Cheng Sun, Hengzhou Xu, Dan Feng, Baoming Bai

Published Sep 1, 2016• 10 citations
Publisher

Abstract

There exist lots of (3, L)-regular quasi-cyclic (QC) LDPC codes constructed from finite fields, protographs, array codes, and computer search under some design rules. For a given code length and rate, how to select the best one from these codes is considerable. In this paper, we study (3, L)-regular QC LDPC codes from the perspective of graph isomorphism, and non-isomorphic (3, L)-regular QC LDPC codes are determined. By analyzing the cycle structures of the resulting non-isomorphic codes, an efficient algorithm for counting cycles is presented. Also proposed is a simplified exhaustive search of non-isomorphic (3, L)-regular QC LDPC codes free of cycles of length less than g0, where g0is the estimated optimal girth value for a given code length. Based on these two algorithms, we can easily construct a (3, L)-regular QC LDPC code with optimized cycle distribution for a given L and code length. Numerical results show that the constructed codes have better performance under the iterative decoding algorithms.

DOI
10.1109/istc.2016.7593119

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There exist lots of (3, L)-regular quasi-cyclic (QC) LDPC codes constructed from finite fields, protographs, array codes, and computer search under some design rules. For a given code length and rate, how to select the best one from these codes is considerable. In this paper, we study (3, L)-regular QC LDPC codes from the perspective of graph isomorphism, and non-isomorphic (3, L)-regular QC LDPC codes are determined. By analyzing the cycle structures of the resulting non-isomorphic codes, an efficient algorithm for counting cycles is presented. Also proposed is a simplified exhaustive search of non-isomorphic (3, L)-regular QC LDPC codes free of cycles of length less than g0, where g0is the estimated optimal girth value for a given code length. Based on these two algorithms, we can easily construct a (3, L)-regular QC LDPC code with optimized cycle distribution for a given L and code length. Numerical results show that the constructed codes have better performance under the iterative decoding algorithms.
There are many algebraic tools, e.g., finite fields and combinatorial designs, for constructing good quasi-cyclic LDPC (QC-LDPC) codes. In particular, QC-LDPC codes designed based on two arbitrary subsets of a given finite field perform very well and some well-known algebraic constructions of QC-LDPC codes have been verified as special cases of this method. How to choose these two subsets to further improve the performance of finite field based QC-LDPC codes is of interest. In this paper, we study such codes and analyze their structural properties from isomorphism perspective, and generalize some rules to efficiently determine the non-isomorphic finite field based codes. By comparing the cycle distributions of non-isomorphic codes, we can easily construct the finite field based QC-LDPC codes with larger girth and fewer short cycles. These codes generally perform very well and can be used as the ingredients of some further processings, e.g., masking. Numerical results show that the constructed codes have better performance with the iterative decoding algorithms.
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.
This paper studies on a class of quasi-cyclic LDPC (QC-LDPC) codes, i.e., Tanner (3, 23)-regular QC-LDPC codes of code length 23pwithpbeing a prime andp≡ 1 (mod 69). We first analyze the cycle structure of Tanner (3, 23)-regular QC-LDPC codes, and divide their cycles of lengths 4, 6, 8, and 10 into five equivalent types. We propose the sufficient and necessary condition for the existence of these five types of cycles, i.e., the polynomial equations in a 69th unit root of the prime field Fp. We check the existence of solutions for such polynomial equations by using the Euclidean division algorithm and obtain the candidate girth values of Tanner (3, 23)-regular QC-LDPC codes. We summarize the results and determine the girth distribution of Tanner (3, 23)-regular QC-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.
There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them are focused on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and also other method is proposed for regular QC LDPC codes with girth at least 14. A construction algorithm of QC LDPC codes by lifting multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes, the progressive edge-growth (PEG) LDPC codes, and the PEG QC LDPC codes.
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes require less hardware resources due to the quasi-cyclic structure of parity-check matrices.In this paper, two improved methods for constructing QC-LDPC codes are proposed.Firstly, in order to make sure the masked LDPC codes have flexible code rate and code length, an improved construction of masking matrix is presented.The proposed regular masking matrix of QC-LDPC codes have the girth at least 6.Then, the irregular codes are constructed by using improved finite field approach and PEG algorithm, which result in large girth and less short cycles.Simulations demonstrate that the proposed QC-LDPC codes have both lower error floor and good waterfall performance.