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IEEE Communications Letters

On the Construction of LDPC Codes Free of Small Trapping Sets by Controlling Cycles

Xiongfei Tao, Yufei Li, Yonghe Liu, Zuoqi Hu

Published Mar 8, 2017• 30 citations
Publisher

Abstract

Low-density parity-check (LDPC) codes exhibit excellent error correcting capability. However, small trapping sets in the Tanner graph are harmful to the iterative decoding algorithm. In this letter, we present a method of constructing (3, n) girth-eight quasi-cyclic LDPC codes with low error floor by removing the small trapping sets from the Tanner graph. To address this issue, we analyze the relationship between eight-cycles and small trapping sets of Tanner graphs based on fully connected base graphs without parallel edges. We find that if some eight-cycles are not found in the Tanner graphs, any elementary trapping set in the range of a ≤ 8 and b ≤ 3 is removed naturally. We also derive a lower bound on the permutation size for the construction of such codes. The experimental simulation shows a favorable error rate performance with lower error floor over additive white Gaussian noise channels.

DOI
10.1109/lcomm.2017.2679707

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Low-density parity-check (LDPC) codes exhibit excellent error correcting capability. However, small trapping sets in the Tanner graph are harmful to the iterative decoding algorithm. In this letter, we present a method of constructing (3, n) girth-eight quasi-cyclic LDPC codes with low error floor by removing the small trapping sets from the Tanner graph. To address this issue, we analyze the relationship between eight-cycles and small trapping sets of Tanner graphs based on fully connected base graphs without parallel edges. We find that if some eight-cycles are not found in the Tanner graphs, any elementary trapping set in the range of a ≤ 8 and b ≤ 3 is removed naturally. We also derive a lower bound on the permutation size for the construction of such codes. The experimental simulation shows a favorable error rate performance with lower error floor over additive white Gaussian noise channels.
We present a method to construct low-density parity-check (LDPC) codes with low error floors on the binary symmetric channel. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the trapping set ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum–product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four cycles.
Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ξ of lengthl(ξ) in the base graph such that the inverse image of ξ in the lifted graph consists of only cycles of length strictly larger thanl(ξ). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, and in some cases, the code rate. The constructed codes are quasi-cyclic and thus attractive from a practical point of view. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.
One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An (a,b)(a,b)(a,b) elementary trapping set, or simply an ETS where aaa is the size and bbb is the number of degree-one check nodes and ba<1\frac{b}{a}<1ab​<1, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected (3,n)(3,n)(3,n)-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs.
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 letter, we propose a new technique to design quasi-cyclic low-density parity-check (QC-LDPC) codes with girth 8 and variable node degrees 3 and 4. Small trapping sets are avoided in the designed codes by imposing simple constraints on the cycles of length 8 in the code's Tanner graph. We also derive lower bounds on the block length of the codes that can satisfy such constraints. Simulation results are provided to demonstrate the superiority of the designed codes in comparison with similar codes in the literature.
Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ccc of length ℓ(c)\ell(c)ℓ(c) in the base graph such that the inverse image of ccc in the lifted graph consists of only cycles of length strictly larger than ℓ(c)\ell(c)ℓ(c). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, encoder and decoder structure, and in some cases, the code rate. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.