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Journal of Hangzhou Dianzi University

The Design of Quasi-cyclic LDPC Codes Based on Cyclic Difference Sets

Zhao Ze-mao

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Combining the perfect cyclic difference sets with the quasi-cyclic codes,a new algorithm for constructing QC-LDPC codes based on cyclic difference sets is proposed.First,the author obtains a small initial matrix that meet the requirements of the weights based on cyclic difference sets,and then the zeros elements,non-zero elements of the initial matrix are replaced by all-zero matrices and cyclic shift matrices.For the purpose of determining the shift coefficient,the cycle length of six is eliminated by computer searching.Simulations show that,under the AWGN channel and the BP decoding algorithm,our QC-LDPC codes are better than the random LDPC codes in the bit error rate and they are more proper for hardware implementation.

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The existing construction of Quasi-Cyclic low-density parity-check (short for QC-LDPC) codes has not considered the problems of small stopping sets and small girth and small minimum code weight together, while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the QC-LDPC codes without small stopping sets and small girth were proposed first. Then good QC-LDPC codes without small stopping sets and small girth were designed based on the row vectors of generator matrix. The small minimum code weight also can be prevented due to the new algorithm based on the row vectors of generator matrix, which is the new way to choose the shifted factors to construct the good QC-LDPC codes for the given prime. The effectiveness and the practicability of the algorithm are demonstrated by the simulation results. It is significant for us to analyze and optimize the design of LDPC codes.

IET International Conference on Wireless, Mobile and Multimedia Networks (ICWMMN)•Jan 1, 2008•1 citation
Combining the perfect cyclic difference sets with the quasi-cyclic codes,a new algorithm for constructing QC-LDPC codes based on cyclic difference sets is proposed.First,the author obtains a small initial matrix that meet the requirements of the weights based on cyclic difference sets,and then the zeros elements,non-zero elements of the initial matrix are replaced by all-zero matrices and cyclic shift matrices.For the purpose of determining the shift coefficient,the cycle length of six is eliminated by computer searching.Simulations show that,under the AWGN channel and the BP decoding algorithm,our QC-LDPC codes are better than the random LDPC codes in the bit error rate and they are more proper for hardware implementation.
Based on the characteristics of optical orthogonal codes,a new algorithm for constructing quasi-cyclic LDPC codes is proposed.Firstly,based on optical orthogonal codes,we can get an initial matrix which meets the parameter request,then under the proposed formula and the characteristics of quasi-cyclic LDPC codes to determine the shift parameter matrix.Finally,using all-zero matrix and the circlet permutation matrix to fill the shift parameter matrix,we can get a check matrix without cycles with length of 4 and 6.Simulations show that,under the AWGN channel and the BP decoding algorithm,the new quasi-cyclic LDPC codes are better than Mackay's codes in the bit error rate and they are more proper for hardware implementation.
In order to solve the problems of high coding complexity and inflexible code length selection in quasi-cyclic low-density Parity-check (QC-LDPC) codes, a deterministic structure is proposed based on the perfect cyclic difference sets. The number of shifts of base matrix can be obtained by a simple addition and subtraction. Because of the special structure of the base matrix and the perfect cyclic difference sets, it saves the storage space and reduces the complexity of the hardware implementation. The girth is at least six and the code-length and code-rate can be selected flexibly. Simulation results demonstrate that the Net Coding Gain of the irregular PCDS-QC-LDPC(2680, 1340) code based on perfect cyclic difference sets has respectively improved 0.13dB and 0.32dB than the code based on PEG-QC-LDPC(2680,1340) and array dispersion and masking AD-MASK-QC-LDPC(2680,1340) with the rate of 0.5 and the bit error rate of 10 -6 by using the sum-product algorithm iterative decoding in the additive white Gaussian channel.
Quasi-cyclic (QC) Low-density paritycheck (LDPC) codes are constructed from combination of weight-0 (null matrix) and Weight-2 (W2) Circulant matrix (CM), which can be seen as a special case of the general type-II QC LDPC codes. The shift matrix of the codes is built on the basis of one integer sequence, called perfect Cyclic difference set (CDS), which guarantees the girth of the code at least six. Simulation results show that the codes can perform well in comparison with a variety of other LDPC codes. They have excellent error floor and decoding convergence characteristics.
For the issue of determining cyclic shift coefficients of the quasi-cyclic sub-matrix in the quasi-cyclic low-density parity-check, a method was presented based on Yang Hui triangle to compute the cyclic shift coefficients. By this method, cyclic shift coefficients could be expressed in simple analytic expressions, and cycles of length four in parity matrix were eliminated. The parity matrix is quasi-cyclic to save required memory and is prone to coding and decoding, and has high flexibility with respect to the design of code length and rate. Over an additive white Gauss noise channel and under the BP decoding algorithm, simulations show that the SNR of the QC-LDPC codes with the proposed algorithm is better than random codes close to 0.3 dB at the BER performance of 10-4 and better than the LDPC codes in DVB-S2 0.2 dB at the BER performance of 10-6. Moreover, the BER performance of the new codes is no less than the LDPC codes in IEEE 802.16e under the same conditions. Furthermore, the simulation result also indicates that by making a reasonable choice of the size of the cyclic shift matrix, the BER performance of the codeword can be improved.
This paper presents an algebra method for constructing LDPC code based on Vandermonde matrix,which includes quasi-cyclic Q matrix and the perfect cyclic difference sets.Employing quasi-cyclic Q matrix as sub-matrix,and Vandermonde matrix as mother matrix,finally gets the H matrix.The quasi-cyclic Q matrix has a high flexibility concerning design of the code rate and the code length,which makes it compatible with the existed standards.The combinatorial simulation results from Matlab and C show that the complexity of this algorithm for encodeing is O(N) and relatively easy to implement.By using this method,we get low bit-error-rate with about 10-8 @ 0.5 code rate under the AWGN channel and don't detect the error floor in decoding.
The existing construction of Quasi-Cyclic low-density parity-check (short for QC-LDPC) codes has not considered the problems of small stopping sets and small girth and small minimum code weight together, while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the QC-LDPC codes without small stopping sets and small girth were proposed first. Then good QC-LDPC codes without small stopping sets and small girth were designed based on the row vectors of generator matrix. The small minimum code weight also can be prevented due to the new algorithm based on the row vectors of generator matrix, which is the new way to choose the shifted factors to construct the good QC-LDPC codes for the given prime. The effectiveness and the practicability of the algorithm are demonstrated by the simulation results. It is significant for us to analyze and optimize the design of LDPC codes.