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Dianzi qijian

Construction of LDPC Code Based on Vandermonde Matrix

Zongsheng Lai

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
A method to construct a flexible length or rate of quasi-cyclic Low Density Parity Check(LDPC) codes based on Balanced Incomplete Block Design(BIBD) and cycle permutation matrix is proposed.Based on actual demand,the corresponding template matrix is constructed by BIBD method.The template matrix is extended by the appropriate cycle permutation matrix.The quasi-cyclic LDPC codes constructed by the proposed method have a good structure and flexibility.This method improves the efficiency of construction good codes which meet the needs of a practical application.Simulations show that quasi-cyclic LDPC codes constructed by the proposed method perform well over AWGN channels with iterative decoding and have a lower error floor.
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 this contribution, we investigate the attainable performance of quasi-cyclic (QC) protograph low-density parity-check (LDPC) codes for transmission over both Additive White Gaussian noise (AWGN) and uncorrelated Rayleigh channels. The codes presented are constructed using the Vandermonde matrix and benefit from both low- complexity encoding and decoding, low memory requirements as well as hardware-friendly implementations. Our simulation results demonstrate that the advantages offered by this family of QC protograph LDPC codes accrue without any compromise in the attainable bit error ratio (BER) and block error ratio (BER) performance. In fact, it is also shown that despite their implementational benefits, the proposed codes exhibit slight BER/BLER gains when compared to some of their more complex counterparts of the same length.
With the growth of length, randomLDPC (lowdensity parity check) codes need large memory to storage the matrixes and have high encoding complexity. In this paper, in order to solve the problems, the algebraic meth- ods based on finite geometry are researched. The parity check matrixes are constructed by the points and lines of fi- nite geometries. The technique of matrix decomposition is used to get irregular QC-LDPC codes with various rates and code lengths. These codes are quasi-cyclic codes and can be encoded with low complexity with a linear rela- tionship to code length. They also have Tanner graphs free of4-cycles. The simulation results indicate: in the con- dition of MSK modulation and AWGN channel, these codes have decoding performance gains about 0.05 to 0.15 dB when bit error rate is 10-6compared with random LDPC codes with similar parameters.