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International Conference on Wireless Communications, Networking and Mobile Computing

A Compact Construction for Nonbinary LDPC Codes Using Permutation Polynomials

Tao Xiong, Hongyu Zhao

Published Sep 1, 2012• 1 citation
Publisher

Abstract

A compact construction for nonbinary low-density parity-check (LDPC) codes over GF(q) (q>2) using permutation polynomials (PPs) is proposed in this paper. Following the previous compact construction for binary LDPC codes using quadratic permutation polynomials (QPPs) over integer rings, a QPP whose coefficients are chosen for maximizing the girth is used for determining all the positions of nonzero elements in the parity check matrix of a regular nonbinary LDPC code. Moreover, it is proposed to use a linear permutation polynomial (LPP) over GF(q) for determining the distribution of nonzero elements of GF(q) at the nonzero positions. Computer simulation results of LDPC codes over GF(8) have shown that the propose compact construction can attain similar error correction performance with Mackay's random construction for nonbinary LDPC codes.

DOI
10.1109/wicom.2012.6478269

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A compact construction for nonbinary low-density parity-check (LDPC) codes over GF(q) (q>2) using permutation polynomials (PPs) is proposed in this paper. Following the previous compact construction for binary LDPC codes using quadratic permutation polynomials (QPPs) over integer rings, a QPP whose coefficients are chosen for maximizing the girth is used for determining all the positions of nonzero elements in the parity check matrix of a regular nonbinary LDPC code. Moreover, it is proposed to use a linear permutation polynomial (LPP) over GF(q) for determining the distribution of nonzero elements of GF(q) at the nonzero positions. Computer simulation results of LDPC codes over GF(8) have shown that the propose compact construction can attain similar error correction performance with Mackay's random construction for nonbinary LDPC codes.
A new block low-density parity-check (Block-LDPC) code based on quadratic permutation polynomials (QPPs) is proposed. The parity-check matrix of the Block-LDPC code is composed of a group of permutation submatrices that correspond to QPPs. The scheme provides a large range of implementable LDPC codes. Indeed, the most popular quasi-cyclic LDPC (QC-LDPC) codes are just a subset of this scheme. Simulation results indicate that the proposed scheme can offer similar error performance and implementation complexity as the popular QC-LDPC codes.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are one of the most important code classes of LDPC codes. They have two drawbacks: lack of randomness and limited girth lead to a degraded decoding performance in the waterfall and error floor regions, respectively. To tackle these problems, we present a new class of LDPC codes, named linear permutation polynomial (LPP) codes, whose parity-check matrix consists of permutation matrices based on LPPs. The girth of regular QC-LDPC codes is upper bounded by 12, while LPP codes break this limit. We demonstrate that LPP codes have error performance almost equivalent to random ones.
A construction is proposed for low density parity check codes using quadratic permutation polynomials over finite integer rings. Graph isomorphisms and automorphisms are identified and used in an efficient search for good codes. Graphs with girth as large as 12 were found. Upper bounds on the minimum Hamming distance are found algorithmically. The bounds indicate that the minimum distance grows with block length. One of the new codes has a similar error performance as the best known PEG LDPC code. Finally, the new codes are quasi-cyclic
A new construction is proposed for low density parity check (LDPC) codes using quadratic permutation polynomials over finite integer rings. The associated graphs for the new codes have both algebraic and pseudo-random nature, and the new codes are quasi-cyclic. Graph isomorphisms and automorphisms are identified and used in an efficient search for good codes. Graphs with girth as large as 12 were found. Upper bounds on the minimum Hamming distance are found both analytically and algorithmically. The bounds indicate that the minimum distance grows with block length. Near-codewords are one of the causes for error floors in LDPC codes; the new construction provides a good framework for studying near-codewords in LDPC codes. Nine example codes are given, and computer simulation results show the excellent error performance of these codes. Finally, connections are made between this new LDPC construction and turbo codes using interleavers generated by quadratic permutation polynomials.
Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q) are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems.The nonbinary codes can achieve significantly better performance for short and moderate block lengths.A lot of works concerning "good" LDPC codes parity check matrix construction has been published so far.However, it is well known that efficient partially parallel hardware decoder architectures are allowed only for codes with blockwise partitioned structure of the parity check matrix, called structured codes.In this paper we present a versatile algorithm for construction of codes that are both nonbinary and structured.The proposed algorithm aims at optimizing the code graph (Tanner graph) by reducing the existence of small cycles with low external connectivity, while at the same time selecting appropriate nonzero coefficients from the Galois Field under interest.The algorithm can be used for code construction of any field order, block length and code rate.
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.