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Telecommunication Engineering

Construction of Nonbinary LDPC Codes Based on BIBD Circulant Permutation Matrices

Jun Gao

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The algebraic method for constructing nonbinary quasi-cyclic(QC) low-density parity-check(LDPC) codes based on balanced incomplete block designs(BIBD) is presented.The generalized nonbinary location vector is proposed on constructions.A nonbinary matrix,which consists of nonbinary circulant submatrices,is formed by generalized two-dimensional matrix dispersion considering the generalized nonbinary location vector and BIBD method.The codes constructed by this method have girths at least 6.Experimental results show that significant coding gains are achieved over Reed-Solomon codes of the same parameters under the Additive White Gaussian Noise(AWGN) channel with iterative decoding Fast Fourier Transform based q-ary Sum-Product Algorithm(FFT-QSPA).And a good performance is also achieved over nonbinary Mackay LDPC codes with almost the same conditions.

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IEEE Transactions on Communications•May 1, 2013•8 citations
The algebraic method for constructing nonbinary quasi-cyclic(QC) low-density parity-check(LDPC) codes based on balanced incomplete block designs(BIBD) is presented.The generalized nonbinary location vector is proposed on constructions.A nonbinary matrix,which consists of nonbinary circulant submatrices,is formed by generalized two-dimensional matrix dispersion considering the generalized nonbinary location vector and BIBD method.The codes constructed by this method have girths at least 6.Experimental results show that significant coding gains are achieved over Reed-Solomon codes of the same parameters under the Additive White Gaussian Noise(AWGN) channel with iterative decoding Fast Fourier Transform based q-ary Sum-Product Algorithm(FFT-QSPA).And a good performance is also achieved over nonbinary Mackay LDPC codes with almost the same conditions.
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.
In this paper, we construct nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes whose parity check matrices consist of an array of square sub-matrices which are either zero matrices or circulant permutation matrices. We propose a novel method to design the shift offset values of the circulant permutation sub-matrices, so that the code length can vary while maintaining a large girth. Extensive Monte Carlo simulations demonstrate that the obtained codes of a wide range of rates (from 1/2 to 8/9) with length from 1000 to 10000 bits have very good performance over both AWGN and Rayleigh fading channels. Furthermore, the proposed method is extended to design multiple nonbinary QC-LDPC codes simultaneously where each individual code can achieve large girth with variable lengths. The proposed codes are appealing to practical adaptive systems where the block length and code rate need to be adaptively adjusted depending on traffic characteristics and channel conditions.
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.
A new combinatorial technique for constructing girth-6 structured binary regular low-density parity-check (LDPC) codes based on special types of t-designs is given. A very large number of well-known t-designs can be used by this method for code construction. Based on this method, a t-(v,k,λ) design D=(X,B) can be exploited for code construction if it satisfies the following three conditions: 1) |B1 ∩ B2|≤ t for any two blocks B1,B2∈ B and B1≠ B2; 2) λ>1; and 3) k>t. Though the technique works for any t-design satisfying these conditions, we focus only on the utilization of simple triple systems, super-simple BIBDs, Steiner systems, and large sets (LSs) of t-designs. We also construct binary and non-binary girth-6 QC-LDPC codes from the t-designs satisfying these conditions by using matrix dispersion method. Experimental results show that the constructed non-binary QC-LDPC codes can provide good practical performance under iterative decoding using the fast Fourier transform based q-ary sum-product algorithm (FFT-QSPA) and they can achieve acceptable coding gains over random-like codes of comparable parameters decoded with sum-product algorithm (SPA).