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Décodage des codes LDPC non binaires : un algorithme à très faible complexité

Adrian Voicila, David Declercq, M.P.C. Fossorier, François S. Verdier, Pascal Urard

Published Jan 1, 2007• 2 citations
OpenAlex

Abstract

In this paper, we introduce a simplified log-domain decoding algorithm for LDPC codes over GF(q), named EMS. While this algorithm is a simplification of the conventional sum-product algorithm , the log-domain decoding has advantages in terms of implementation, computation complexity and numerical stability. Further, we propose some variants of implementation for the EMS algoritm, yielding a lower computational complexity. The different variants are compared with their binary homologous, the Min-Sum algorithm, both in terms of simulated FER performance and computational complexity.

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In this paper, we propose a new implementation of the Extended Min-Sum (EMS) decoder for non-binary LDPC codes. A particularity of the new algorithm is that it takes into accounts the memory problem of the non-binary LDPC decoders, together with a significant complexity reduction per decoding iteration. The key feature of our decoder is to truncate the vector messages of the decoder to a limited number nmof values in order to reduce the memory requirements. Using the truncated messages, we propose an efficient implementation of the EMS decoder which reduces the order of complexity to ¿(nmlog2nm). This complexity starts to be reasonable enough to compete with binary decoders. The performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction, which is shown both with a simulated density evolution approach and actual simulations.

IEEE Transactions on Communications•May 1, 2010•223 citations
In this paper, we introduce a simplified log-domain decoding algorithm for LDPC codes over GF(q), named EMS. While this algorithm is a simplification of the conventional sum-product algorithm , the log-domain decoding has advantages in terms of implementation, computation complexity and numerical stability. Further, we propose some variants of implementation for the EMS algoritm, yielding a lower computational complexity. The different variants are compared with their binary homologous, the Min-Sum algorithm, both in terms of simulated FER performance and computational complexity.
This paper introduces a log-domain decoding scheme for LDPC codes over GF(q). While this scheme is mathematically equivalent to the conventional sum-product decoder, log-domain decoding has advantages in terms of implementation, computational complexity and numerical stability. Further, a suboptimal variant of the log-domain decoding algorithm is proposed, yielding a lower computational complexity. The proposed algorithms and the sum-product algorithm are compared both in terms of simulated BER performance and computational complexity.
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
Low-Density Parity-Check (LDPC) codes have gained significant popularity recently and have been recommended for various wireless standards of the future. Non-binary LDPC codes provide all the well-known advantages of binary LDPC codes but allow for much shorter block lengths. The sum-product decoding algorithm for non-binary LDPC codes is too complex for efficient implementation. Hence, we have represented the sum-product algorithm in a novel log-domain form in terms of operators defined earlier in literature. This form greatly simplifies the key step in the check-node processing and allows us to replace a max* operation in the decoder by a max operation. This leads to a sub-optimal decoding algorithm which we propose in this paper.
Immediate generalization of Gallager's binary low-density parity-check (LDPC) codes gave birth to the LDPC codes over GF(q), which demonstrated error performance exceeding that of binary codes with compromised complexity. Various methods had been proposed for the decoder with an aim to simplifying the belief propagation algorithm. One of the algorithms was the "extended min-sum (EMS)" decoder which concerned only a limited number of incoming messages at the checks. In this paper, we propose a modification to further simplify the EMS decoder without sacrificing performance.
An improved log domain decoding algorithm of Low density parity check (LDPC) codes over GF (q) using permutation to simplify the parity check equation is presented in this paper. This approach is different from the conventional log domain decoding algorithm of Low Density Parity Check (LDPC) codes over GF (q). The difference between improved log domain decoding and conventional log domain is that in improved log domain decoding permutation is applied in check node process where permutation is applied in between check node process and variable node process for conventional log domain decoding. Improved Log domain is mathematically equivalent to the conventional log domain decoding, however improved log-domain has advantages in terms of implementation, computational complexity and numerical stability. The proposed algorithm and the conventional log domain decoding algorithm are compared both in terms of memory requirement and simulated BER performance of (1008, 504) regular LDPC codes over GF (4) having row weight 3 & column weight 6, BPSK modulation.
In this paper, we propose a new implementation of the Extended Min-Sum (EMS) decoder for non-binary LDPC codes. A particularity of the new algorithm is that it takes into accounts the memory problem of the non-binary LDPC decoders, together with a significant complexity reduction per decoding iteration. The key feature of our decoder is to truncate the vector messages of the decoder to a limited number nmof values in order to reduce the memory requirements. Using the truncated messages, we propose an efficient implementation of the EMS decoder which reduces the order of complexity to ¿(nmlog2nm). This complexity starts to be reasonable enough to compete with binary decoders. The performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction, which is shown both with a simulated density evolution approach and actual simulations.