A method for estimating the block error probability of Reed-Muller codes on a binary erasure channel and a binary symmetric channel under a permutation decoding algorithm is proposed. This method is based on an analysis of the block error probability under a recursive decoding algorithm for a fixed number of input erasures or errors. Simulation results demonstrate that the proposed method allows to effectively predict the block error rate performance of Reed-Muller codes under permutation decoding with a different number of permutations used.
A method for estimating the block error probability of Reed-Muller codes on a binary erasure channel and a binary symmetric channel under a permutation decoding algorithm is proposed. This method is based on an analysis of the block error probability under a recursive decoding algorithm for a fixed number of input erasures or errors. Simulation results demonstrate that the proposed method allows to effectively predict the block error rate performance of Reed-Muller codes under permutation decoding with a different number of permutations used.
In this paper, we consider the problem of decoding Reed-Muller (RM) codes in binary erasure channel. We propose a novel algorithm, which exploits several techniques, such as list recursive (successive cancellation) decoding based on Plotkin decomposition, permutations of encoding factor graph as well as the properties of erasure channels.We show that with properly selected number of random permutations, this algorithm considerably outperforms straightforward list decoding while maintaining the same asymptotic complexity. This also means that near-MAP decoding can be achieved with lower complexity cost.
Error exponents are studied for recursive decoding of Reed-Muller (RM) codes and their subcodes used on a binary-symmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplications of the randomly corrupted channel outputs plusmn1 are performed using a specific order of these two operations in each step. Recalculated random outputs are compared in terms of their exponential moments. As a result, tight analytical bounds are obtained for decoding error probability of the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results. Comparison of these bounds with similar bounds for bounded distance decoding and majority decoding shows that recursive decoding can reduce the output error probability of the latter two algorithms by five or more orders of magnitude even on the short block length of 256. It is also proven that the error probability of recursive decoding can be exponentially reduced by eliminating one or a few information bits from the original RM code
Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0,1}m. We denote this code by RM(m,r). The minimal distance of RM(m,r) is 2m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length n=2m) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree r=o(m)) we can correct a random set of (1/2−o(1))n errors with high probability. For high rate codes (of degree m−r for r=o(m/logm)), we can correct roughly mr/2 errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m,m−(2r+2)) for which the same erasure pattern can be corrected in RM(m,m−(r+1)). The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.
Reed-Muller (RM) codes encode an m-variate polynomial of degree at most r by evaluating it on all points in {0,1}m. We denote this code by RM(r,m). The minimum distance of RM(r,m) is 2m-rand so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length n=2m) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree r=o(√m)), we can correct a random set of (1/2-o(1))n errors with high probability. For high rate codes (of degree m-r for r=o(√m/log m)), we can correct roughly mr/2errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m-(2r+2),m), for which the same erasure pattern can be corrected in RM(m-(r+1),m). The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kötter that generalizes the Berlekamp-Welch algorithm for decoding Reed-Solomon codes.
For Reed-Muller (RM) codes of length n and distance d, a recursive decoding algorithm is designed that has complexity of order nlogn for any fixed rate R, and corrects most error patterns of weight up to (dlnd)/2 on the binary symmetric channels (BSC) and up to (2dlnd)/pi on the AWGN channels. For long RM codes of fixed order r, a vanishing decoding error probability and polylogarithmic decoding complexity of order (logn)r+1are obtained on the BSC given any transition probability p that is bounded away from 1/2
Recursive decoding is studied for Reed-Muller (RM) codes used on a binary symmetric channel. Decoding is performed beyond the bounded distance radius d/2 and corrects most error patterns of weight up to (dlnd)/2. In our analysis, decoding is decomposed into consecutive steps, with one information bit derived in each step. Then the error probability of each step is defined by the recursive recalculations of the Bernoulli random variables. We derive the exponential moments of the recalculated random variables. As a result, tight exponential bounds on the output error probability are obtained for the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results