A novel construction method of Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code is proposed in this paper, which is based on protograph and Balanced Block Design (BIBD) in combinational mathematics.The method is to fill the base matrix of protograph with the base set elements of BIBD in accordance with certain mathematical rules.And the base matrix of protograph proposed in the paper has special double diagonal structure, which overcomes the low weight code words in general structure and achieves rapid encoding at the same time.The QC-LDPC code in this paper possesses lower encoding complexity and less memory occupation; therefore it is easy for hardware implementation.In addition, it effectively avoids girth 4. Simulation results show that when the Bit Error Rate (BER) is 10 -7 , QC-LDPC codes based on Protograph and BIBD (PBIBD-LDPC) proposed in this paper has 0.5dB Net Coding Gain (NCG) compared with the protograph QC-LDPC codes that based on Vander monde matrix and PEG algorithm structure, and has 0.15dB NCG compared with the excellent code QC-LDPC that based on the lifting structure.
A novel construction method of Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code is proposed in this paper, which is based on protograph and Balanced Block Design (BIBD) in combinational mathematics.The method is to fill the base matrix of protograph with the base set elements of BIBD in accordance with certain mathematical rules.And the base matrix of protograph proposed in the paper has special double diagonal structure, which overcomes the low weight code words in general structure and achieves rapid encoding at the same time.The QC-LDPC code in this paper possesses lower encoding complexity and less memory occupation; therefore it is easy for hardware implementation.In addition, it effectively avoids girth 4. Simulation results show that when the Bit Error Rate (BER) is 10 -7 , QC-LDPC codes based on Protograph and BIBD (PBIBD-LDPC) proposed in this paper has 0.5dB Net Coding Gain (NCG) compared with the protograph QC-LDPC codes that based on Vander monde matrix and PEG algorithm structure, and has 0.15dB NCG compared with the excellent code QC-LDPC that based on the lifting structure.
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.
To obtain efficient and pragmatic channel codes high signal-to-noise ratios (SNRs), an optimized construction of short and high rate protograph quasi-cyclic (QC) low-density parity-check (LDPC) codes is presented. Based on an optimized protograph template, the code framework is firstly produced by the extensions of the variable nodes. By enlarging the dimension of the sub-matrices related to the protograph framework, the base quasi-cyclic (QC) matrix is generated with required code rate and length. Then, the elements in the base matrix are split with even smaller square sub-matrices of the same row and column weights. In this procedure, a progressive edge growth (PEG) algorithm is employed to find the optimized positions of the QC submatrices, which brings larger girth for better code performance. Also the circulant offsets in all QC sub-matrices are optimally searched by a QC improved approximated cycle extrinsic message degree (QCI-ACE) algorithm, which improves the relationship of the unavoidable loops in the code's Tanner graph and thus it improves the error floor. Simulation results show that the codes produced by the proposed method demonstrate quite good bit error rate (BER) performance. In addition, they exhibit good properties of high spectrum efficiency by the high code rate, as well as the low complexity by the short code length. Therefore, the proposed code construction can be efficiently applied in the design of short and high rate protograph QC-LDPC codes with high rate and rational complexity.
In order to meet the requirements of the high bit-rate and the lower bit error-rate of the errorcorrection code in optical communication systems,based on the balanced incomplete block design(BIBD) and the circulant decomposition in this paper,a novel construction method of quasi-cyclic low-density parity-check(QC-LDPC)codes with the advantage of the simple implementation is proposed.A regular BIBDdes-QC-LDPC(6736,6316)code suitable for optical communication systems is constructed by applying the proposed method.The simulation results show that at the bit error rate(BER)of 10-6 and the same code rate of 93.7%,the net coding gain(NCG)of the constructed BIBDdes-QC-LDPC(6736,6316)code is 2.2dB more than that of the classic RS(255,239)code which has been widely used in optical communication systems,and 0.2dB more than that of the BIBD-QC-LDPC(6736,6315)code with the same code length and the same code-rate constructed only by the BIBD method and that of the irregular QC-LDPC(3843,3603)code with the same code-rate based on Galois field multiplicative group.Therefore,the novel BIBDdes-QC-LDPC(6736,6316)code has superior error-correction performance and is more suitable for the high-speed long-haul optical communication systems.And the novel construction method still has the advantage of the BIBD method which can flexibly adjust the code length and code rate.
Based on the Balanced Incomplete Block Design(BIBD),a novel construction method of Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes is further analyzed and researched.And the three QC-LDPC codes with the same code-rate and different code-length are constructed by applying the proposed method.The simulation analysis shows that under the condition of the same code-rate,the QC-LDPC code with the longer code-length has better error-correction performance.Simultaneously,at the condition of the bit error rate(BER) of 10-6 and the same code rate of 93.7%,the net coding gain(NCG) of the constructed BIBD-QC-LDPC(5392,5056) code is respectively 2.13 dB and 1.41 dB higher than that of the classic RS(255,239) code widely used in optical communication systems and that of the LDPC(32640,30592) code in ITU-T G.975.1.Therefore,the BIBD-QC-LDPC(5392,5056) code has much superior error-correction performance and is better suitable for high-speed long-haul optical communication systems.
For a given base graph, the lifted graph can be obtained by a copy-and-permute procedure. If the permutation is cyclic, the lifted graph corresponds to a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code. The girth of the QC protograph LDPC code is determined by the girth of the base graph and the permutation shifts. In this paper, we first derive a lower bound on the lifting degree to achieve a large girth lifted graph. Then, motivated by the cycle searching and girth maximizing features of the progressive edge-growth (PEG) algorithm, we introduce the permutation shifts determining (PSD) PEG algorithm, which can construct large girth base graph and determine the optimal permutation shifts, simultaneously. It is shown that the computational complexity of PSD-PEG algorithm is much lower than that of the PEG algorithm and the PEG-QC algorithm for the same codeword length. Furthermore, we show that the PSD-PEG algorithm can also be used to construct nonbinary QC protograph LDPC codes without low weight codes. Simulation results show that the binary and nonbinary QC protograph LDPC codes constructed by the PSD-PEG algorithm have good bit error rate performance and frame error rate performance over the additive white Gaussian noise channel.
Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting. However, these restrictions impose undesirable fixed upper limits on important code parameters, such as minimum distance and girth. In this paper, we consider an approach to constructing QC-LDPC codes that uses a two-step lifting procedure based on a protograph, and, by following this method instead of the usual one-step procedure, we obtain improved minimum distance and girth properties. We also present two new design rules for constructing good QC-LDPC codes using this two-step lifting procedure, and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared to one-step circulant-based liftings. The expected performance improvement is verified by simulation results.