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IET Communications

Modified majority logic decoding of Reed–Muller codes using factor graphs

Ting‐Ya Yang, Houshou Chen

Published Jan 31, 2018• 2 citations
Publisher

Abstract

This study investigates the decoding algorithms of binary Reed–Muller (RM) codes. The main goal of the study is to modify majority‐logic decoding using sum‐product or min‐sum decoding algorithms at the highest‐order information bits in RM codes. Numerical results show that compared with the traditional majority‐logic decoding algorithm, the modified majority‐logic decoding method proposed in this study can reduce the bit error rate by a large margin. This is achieved without incurring a significant increase in decoding complexity because soft‐decision decoding is only used in the highest information bits in RM codes whereas majority‐logic decoding is still applied at the other orders of information bits in RM codes.

DOI
10.1049/iet-com.2017.0987

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IEEE Transactions on Information Theory•Mar 1, 2006•163 citations
This study investigates the decoding algorithms of binary Reed–Muller (RM) codes. The main goal of the study is to modify majority‐logic decoding using sum‐product or min‐sum decoding algorithms at the highest‐order information bits in RM codes. Numerical results show that compared with the traditional majority‐logic decoding algorithm, the modified majority‐logic decoding method proposed in this study can reduce the bit error rate by a large margin. This is achieved without incurring a significant increase in decoding complexity because soft‐decision decoding is only used in the highest information bits in RM codes whereas majority‐logic decoding is still applied at the other orders of information bits in RM codes.
This paper proposes the modify majority-logic decoding using sum-product or min-sum decoding algorithms at the highest-order information bits in RM codes. Numerical results show that, compared with the traditional majority-logic decoding algorithm, the modified majority logic decoding method proposed in this study can reduce the bit error rate (BER) considerably. This task is accomplished without incurring a significant increase in decoding complexity because soft-decision decoding is used only at the highest information bits of RM codes, whereas majority-logic decoding is still applied at the other orders of information bits.
In this paper, novel hard-decision iterative decoding algorithms for binary Reed-Muller (RM) codes are presented. First, two algorithms are devised based on the majority-logic decoding algorithm with reliability measures of the received sequence. The bit-flipping (BF) and the normalized bit-flipping (NBF) decoding algorithms are hard-decision decoding algorithms. According to the updated hard reliability measures, the BF and NBF algorithms flip one bit of the received hard-decision sequence at a time in each iteration. The NBF decoding algorithm performs better than the BF decoding algorithm by normalizing the reliability measures of the information bits. Moreover, the BF and NBF algorithms are modified to flip multiple bits in one iteration to reduce the average number of iterations. The modified decoding algorithms are called the multiple-bits-flipping (MBF) algorithm and the normalized multiple-bits-flipping (NMBF) algorithm, respectively. The proposed algorithms have low computational complexities and can converge rapidly after a small number of iterations. The simulation results show that the proposed hard-decision decoding algorithms outperform the conventional decoding algorithm.
The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.
We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.
We present a new soft decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of all received symbols are recalculated into the reliabilities of the parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to the more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft decision algorithm outperforms its conventional hard decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output bit error rate /spl epsi/<1/2.
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes