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IEEE Transactions on Communications

Generalized Binary Representation for the Nonbinary LDPC Code With Decoder Design

Yang Yu, Wen Chen, Jun Li, Xiao Ma, Baoming Bai

Published Jul 30, 2014• 6 citations
Publisher

Abstract

In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e.,$\bar{\cal C}$. A new methodology for constructing the binary representation [generalized binary representation (GBR)] of$\bar{\cal C}$is proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother code$\bar{\cal C}$under nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.

DOI
10.1109/tcomm.2014.2344912

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In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e.,Cˉ\bar{\cal C}Cˉ. A new methodology for constructing the binary representation [generalized binary representation (GBR)] ofCˉ\bar{\cal C}Cˉis proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother codeCˉ\bar{\cal C}Cˉunder nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.
The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.
The equivalent binary parity check matrices for the binary images of the cycle-free non-binary LDPC codes have numerous bit-level cycles. In this paper, we show how to transform these binary parity check matrices into their cycle-free forms. It is shown that the proposed methodology can be adopted not only for the binary images of non-binary LDPC codes but also for a large class of binary LDPC codes. Specifically, we present an extended ppp-reducible (EPR) LDPC code structure to eliminate the bit-level cycles. For the non-binary LDPC codes with short length symbol-level cycles, the EPR-LDPC codes can largely avoid the corresponding short length bit-level cycles. As to the decoding of the EPR-LDPC codes, we propose a hybrid hard-decision decoder and a hybrid parallel decoder for binary symmetric channel and binary input Gaussian channel, respectively. A simple code optimization algorithm for these binary decoders is also provided. Simulations show the comparative results and justify the advantages, i.e., better performance and lower decoding complexity, of the proposed binary constructions.
Low-density parity-check (LDPC) codes over finite fields GF(q) have a better performance than those of the binary low-density parity-check codes, at short and medium block lengths, but the decoder of GF(q)-LDPC has more complexity. In this paper, we present a new reduced-complexity belief propagation (BP) decoding algorithm for GF(q)-LDPC codes by analyzing the reliability of code symbols. Instead of update all the variable nodes during iteration, we just update variable nodes which are mostly to be in an error. The algorithm also shows a property to reduce the effects of oscillating on decoding. The simulations prove that we could improve performance and reduce complexity at the same time for short and medium GF(q)-LDPC codes.
Decoding Algorithms for Nonbinary LDPC Codes Over GF(q)(q)(q)
In this letter, we address the problem of decoding nonbinary low-density parity-check (LDPC) codes over finite fields GF(q), with reasonable complexity and good performance. In the first part of the letter, we recall the original belief propagation (BP) decoding algorithm and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding. In the second part of the letter, we introduce a simplified decoder which is inspired by the min-sum decoder for binary LDPC codes. We called this decoder extended min-sum (EMS). We show that it is possible to greatly reduce the computational complexity of the check-node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance is quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very-high-order fields show that nonbinary LDPC codes are promising for applications which require low frame-error rates for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes
We propose an approach for optimizing nonbinary (NB) quasi-cyclic (QC) LDPC codes. This approach combines constructing of base parity-check matrices by simulated annealing and labeling the obtained base matrices aimed at maximizing the so-called generalized girth of the NB LDPC code Tanner graph. Tightened random coding bounds based on the average binary spectra for ensembles of "almost regular" NB LDPC codes of finite lengths over extensions of the binary Galois field are derived. The simulated FER performance of the sum-product BP decoding of "almost regular" NB QC-LDPC block codes are presented and compared with the derived finite-length random coding bounds as well as with the same performance of the optimized binary QC-LDPC block code in the 5G standard. In the waterfall region our finite-length bounds on the error probability of ML decoding are about 0.1--0.2 dB away from the simulated FER performance of BP decoding.
This paper addresses decoder design for nonbinary quasicyclic low-density parity-check (QC-LDPC) codes. First, a novel decoding algorithm is proposed to eliminate the multiplications over Galois field for check node processing. Then, a partially parallel architecture for check node processing units and an optimized architecture for variable node processing units are developed based on the new decoding algorithm. Thereafter, an efficient decoder structure dedicated to a promising class of high-performance nonbinary QC-LDPC codes is presented for the first time. Moreover, an ASIC implementation for a (620, 310) nonbinary QC-LDPC code decoder over GF(32) is designed to demonstrate the efficiency of the presented techniques.