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IET International Conference on Wireless, Mobile and Multimedia Networks (ICWMMN)

A class of (3, k) quasi-cyclic LDPC codes from difference sequences with girth 8

Bing Li, Lijun Zhang, L. L. Cheng

Published Jan 1, 2011• 3 citations
Publisher

Abstract

An approach for constructing a class of (3, k)-regular quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed, which is based on combinatorial objects termed difference sequences. By an efficient algorithm for searching good difference sequences, codes in this class have girth at least eight. Simulation results show that the codes slightly outperform the counterpart PEG codes and have better performance than the corresponding MacKay codes and array codes.

DOI
10.1049/cp.2011.0969

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An approach for constructing a class of (3, k)-regular quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed, which is based on combinatorial objects termed difference sequences. By an efficient algorithm for searching good difference sequences, codes in this class have girth at least eight. Simulation results show that the codes slightly outperform the counterpart PEG codes and have better performance than the corresponding MacKay codes and array codes.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.
In this paper, we consider the constructions of type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing a girth-eight type-II QC-LDPC code. By combining these conditions with the concept of the Sidon sequence, three classes of type-II QC-LDPC codes are subsequently proposed with girth eight. To the best of our knowledge, the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth-eight type-II QC-LDPC codes that have rates being greater than a half and, at the same time, have distance upper bounds being not limited to 12. Via simulations, we show the promising performance of the proposed type-II QC-LDPC codes with girth eight. In addition, a set of general bounds and explicit/random constructions without using a Sidon sequence are also presented for the type-II QC-LDPC codes with girth six or eight. We observe that the proposed codes perform better than or equally well as the random QC-LDPC codes from PEG and quadr.-congr. methods, while the novel codes possess both highly structured parity-check matrices and very flexible choices in circulant size.
A new class of binary low-density parity-check(LDPC)codes is proposed based on B2(mod m) sequences.The parity-check matrix of such a code has a column weight of three and a row weight of an arbitrary integer,and a quasi-cyclic structure.The parity-check matrix also has a girth at least 8,and corresponds to a code with minimal distance at least 12.When m is prime,an 8-cycles reduction method is presented to completely avoid the two types of 8-cycles within the total four types existed in the Tanner graph.Simulation results show that,for a prime integer m,the new LDPC code outperforms the random(quasi-)regular counterpart generated by the PEG algorithm.Finally,a heuristic algorithm based on a strategy called neighboring extension search is presented to search for the B2(mod m)sequences whose lengths approach or meet the upper bound.
This paper proposes a construction method of regular(j, k) Lucas quasi-cyclic low-density parity-check(L-QC-LDPC) codes with girth at least eight based on the Lucas sequence. By this method, the girth of L-QC-LDPC codes is large, which can effectively eliminate short cycles. Besides, lower bound of circulant permutation submatrix dimension p is allowed continuous values. In addition, it can save the storage space in terms of hardware implementation which reduces the cost and complexity of hardware realization correspondingly. Simulation results show that the L-QC-LDPC codes have net coding gain(NCG) of about 2d B and 0.8d B compared with one-coincidence sequence quasi-cyclic LDPC(OCS-LDPC) codes and deterministic codes, respectively, at code rate of 1/2, code length of 1302 and bit error rate(BER) of 10?6. At the same condition, the performance of L-QC-LDPC codes is slightly better than that of the quadratic function LDPC(QF-LDPC) codes, which has an NCG improvement of around 0.1d B. Meanwhile, at code rate of 1/2, similar code length and BER of 10~(-6), the NCG of L-QC-LDPC codes outweigh about 0.5d B compared with that of QC-LDPC codes based on cyclic subgroups of finite fields.
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.