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Journal of Systems Engineering and Electronics

Ruling out small stopping sets and small girth in Tanner graph of QC-LDPC code

Ling-jun, Kong, Yang Yang, Xiao Xiao

Published Jan 1, 2010• 0 citations
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A new algorithm of counting the number of small stopping sets and girth in QC-LDPC Codes

Lingjun Kong, Yang Xiao

It is well known that the performance of low-density parity check (LDPC) codes under iterative decoding is determined by certain combinatorial structures (such as stopping sets and girth) in the Tanner graph. The difficulty in enumerating all possible combinations of the columns may prevent an efficient search of good LDPC codes with small stopping sets and girth. To solve the problem, this paper presents a new algorithm of counting the number of small stopping sets and girth by analyzing the shapes of the cycles of Tanner graph in parity check matrix for designing good LDPC codes which is less complex than the existing algorithms. This method can be used effectively to evaluate the performance of LDPC codes according to their small stopping sets and girth distributions.

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Design of good QC-LDPC codes without small girth in the p-plane

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A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate (BER) performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.

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QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field

Farzane Amirzade, Mohammad‐Reza Sadeghi, Daniel Panario

Trapping sets significantly influence the performance of low-density parity-check codes. An $ (a, b) $ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $ a $ and $ b $ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $ (3, n) $-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected $ (3, n) $-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of $ (a, b) $ ETSs with $ a\leq5 $ and $ b\leq2 $. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.

Advances in Mathematics of Communications•Jan 1, 2020•10 citations

Design of good QC-LDPC codes without small girth in the p-plane

Ling-jun, Kong, Yang Yang +1 more

Journal of Systems Engineering and Electronics•Jan 1, 2011•0 citations
It is well known that the performance of low-density parity check (LDPC) codes under iterative decoding is determined by certain combinatorial structures (such as stopping sets and girth) in the Tanner graph. The difficulty in enumerating all possible combinations of the columns may prevent an efficient search of good LDPC codes with small stopping sets and girth. To solve the problem, this paper presents a new algorithm of counting the number of small stopping sets and girth by analyzing the shapes of the cycles of Tanner graph in parity check matrix for designing good LDPC codes which is less complex than the existing algorithms. This method can be used effectively to evaluate the performance of LDPC codes according to their small stopping sets and girth distributions.
A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate (BER) performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.
Trapping sets significantly influence the performance of low-density parity-check codes. An $ (a, b) $ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $ a $ and $ b $ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $ (3, n) $-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected $ (3, n) $-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of $ (a, b) $ ETSs with $ a\leq5 $ and $ b\leq2 $. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.