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Dianzi Ke-ji Daxue xuebao

Construction Method of Large Girth QC-LDPC Codes Based on Lucas Sequences

Huang Shen

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

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.

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Space Electronic technology•Jan 1, 2013•0 citations
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.
One approach for designing the one-coincidence sequence (OCS) low-density parity-check (LDPC) codes of large girth is investigated. These OCS-LDPC codes are quasi-cyclic, and their parity-check matrices are composed of circulant permutation matrices. Generally, the cycle structures in these codes are determined by the shift values of circulant permutation matrices, and the existence of cycles in the corresponding Tanner graph is governed by certain cycle-governing equations (CGEs). Therefore, finding the proper shift values is the key point to increase the girth of these codes. In this paper, we provide an effective method to systematically find out the CGEs for these codes of girth 6, 8, and 10, respectively. Then, one less computation-intensive algorithm is used to generate the proper shift values for constructing the OCS-LDPC codes of large girth. Simulation results show that significant gains in signal-to-noise ratio over an additive white-Gaussian noise channel can be achieved by increasing the girth of the OCS-LDPC codes.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
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.
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.
Due to quasi-cyclic structure of parity-check matrices, Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes reduce hardware requirements, and thus they have a higher practical value than random LDPC codes. This paper proposes an improved construction method of QC-LDPC based on PEG Algorithm, and the girth of the QC-LDPC codes has an increase of 2. Firstly, a base matrix is generated by the PEG algorithm, and then the base matrix is expanded using the unit circle permutation matrix, at the same time, the short loop of the base matrix is eliminated. This approach has the advantages of the PEG algorithm and the QC-LDPC codes, and simulation results show that bit error rate (BER) performance of the QC-LDPC codes presented in this paper is the same as or exceeds the one of randomly constructed LDPC codes.
By classifying cycle types within the parity-check matrix of a quasi-cyclic(QC) low-density parity-check(LDPC) code,a method without repetition or omission is proposed to detect all the cycles with lengths smaller than twelve.Based on this cycle detection method,a new approach is presented to construct large girth QC-LDPC codes by using some heuristic strategies.The approach includes three steps.Firstly,under the infinite cyclic permutation matrix(CPM) size condition,two integer sequences satisfying the girth constraint are found using a heuristic strategy.Secondly,according to the designed code rate,several integer pairs are extracted from the two sequences so as to form a shift matrix.Finally,by employing the recently proposed theoretical lower bound by Zhang et.al,a range of consecutive CPM sizes in which the girth never decrease is accurately computed from the shift matrix.Compared with Liu-Han's method,the new approach possesses two advantages in the sense that:(1) Not only girth-10+ QC-LDPC codes but also girth-12 ones can be constructed to have consecutive CPM sizes;(2) girth-10+ and girth-12 QC-LDPC codes can be generated with very high code rates,due to the new cycle detection method which significantly accelerates the sequence search procedure.