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International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS)

A parity check matrix design for Irregular LDPC codes with 2K block length

Chutima Prasartkaew, Somsak Choomchuay

Published Dec 1, 2009• 6 citations
Publisher

Abstract

This paper outlines the work on another design of a parity check matrix for Irregular LDPC codes. The design is based on the pattern of Modified Array and Interleaved Modified Array LDPC codes. The application of matrix transposition Quasi-cyclic shifting has resulted in the reduction of 1's. The designed matrix is suitable for codes with short and medium block lengths. The code rate of 0.56 at the BER of 10-4is obtained.

DOI
10.1109/ispacs.2009.5383779

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This paper outlines the work on another design of a parity check matrix for Irregular LDPC codes. The design is based on the pattern of Modified Array and Interleaved Modified Array LDPC codes. The application of matrix transposition Quasi-cyclic shifting has resulted in the reduction of 1's. The designed matrix is suitable for codes with short and medium block lengths. The code rate of 0.56 at the BER of 10-4is obtained.
The existing design of quasi-cyclic(QC) low-density parity-check(LDPC) codes has not considered the problems of short length girths and the rows' dependency.The first problem leads to the BER performance of QC LDPC codes to be much poorer than that of randomly constructed LDPC codes,while the second problem leads to the difficulty of construction of generator matrices from parity-check matrices.To solve the first problem,the constraint conditions for designing the QC LDPC codes are proposed,and the dimension and shift factors of the circulant matrices of the given sparse parity-check matrix are adjusted according to the test results of both girth 4 and girth 6.To solve the second problem,an approach of irregular QC LDPC codes is proposed.The proposed approach is to replace some sub-matrices by zero matrices and identity matrices at special positions in the given sparse parity-check matrix so as to get a nonsingular square matrix for the construction of the generator matrix.Though adopting bidiagonal submatrices in parity-check matrix can solve the rows' dependency problem,many code words with low weights will occur,which leads to the BER performance can not be better by increasing code lengths.Examples are provided for the proposed design of QC LDPC codes,and computer simulation results show that the proposed QC LDPC codes achieve good BER performance.
This paper presents a construction algorithm for the short block irregular low-density parity-check (LDPC) codes. By applying a magic square theorem as a part of the matrix construction, a newly developed algorithm, the so-called Magic Square Based Algorithm (MSBA), is obtained. The modified array codes are focused on in this study since the reduction of 1s can lead to simple encoding and decoding schemes. Simulation results based on AWGN channels show that with the code rate of 0.8 and SNR 5 dB, the BER of �� �� can
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.
A design algorithm of regular check matrix for Low-Density Parity-Check(LDPC)codes is proposed in this paper.The proposed approach is to design 3 sub-matrices,and each of the sub-matrices is combined by the cyclic shift matrices of the identity matrix,then to combine these sub-matrices into a desired parity check matrix according to our approach.Using the check algorithms of the short cycles to find the combinations of shifting operators by which the parity check matrix desires has no girth- four and no girth-six.The simulation results in AWGN channel show that the BER performance of these codes obtained by the generator matrix derived from the proposed check matrices to encode the random information bits is as good as that of random LDPC codes.
Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.