Efficiently decoding Reed-Muller codes from random errors

Abstract

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 2 m−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 = 2 m ) 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/ log m)), we can correct roughly m r/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 codes achieve capacity on erasure channels

“…In particular, [28,Theorem 1.8] shows that an error pattern can be corrected by RM(n − (2t + 2), n) under block-MAP decoding whenever an erasure pattern with the same support can be corrected by RM(n − (t + 1), n) under block-MAP decoding. Using the algorithm in [88], these error patterns can even be corrected efficiently. Combined with our results for the BEC, [88,Corollary 14] shows that there exists a deterministic algorithm that runs in time at most n 4 and is able to correct (1/2−o(1))2 n random errors in RM(n, o( √ n))…”

“…Using the algorithm in [88], these error patterns can even be corrected efficiently. Combined with our results for the BEC, [88,Corollary 14] shows that there exists a deterministic algorithm that runs in time at most n 4 and is able to correct (1/2−o(1))2 n random errors in RM(n, o( √ n))…”

Sparse Multi-Decoder Recursive Projection Aggregation for Reed-Muller Codes

“…In [12], minimum-weight parity checks are employed taking advantage of the large automorphism group of RM codes. Berlekamp-Welch type algorithms on random errors and erasures are considered and analyzed in [13], [14]. Successive cancellation (SC) decoding [9] and SC list (SCL) decoding [15], initially proposed for polar codes, are also applicable to RM codes as they share a similar construction.…”

Codes approaching the Shannon limit with polynomial complexity per information bit

“…It was proven in [3,4] that long low-rate RM codes RM (m, r) of order r = o(m) and length n = 2 m approach the maximum possible code rates C m on channels W m under the maximumlikelihood (ML) decoding. Even in this case, code rates R n decline exponentially as m r 2 −m and require exponential decoding complexity.…”