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Journal of Communications and Networks

Construction of block-LDPC codes based on quadratic permutation polynomials

Guan Wu, Liping Liang

Published Apr 1, 2015• 7 citations
Publisher

Abstract

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.

DOI
10.1109/jcn.2015.000029

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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.
In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices via a simple quadratic congruential equation. The main advantage is that QC-LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC-LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths and have almost the same performance as Sridara-Fuja-Tanner codes.
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.
In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.
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