Efficiently Decoding Reed-Muller Codes From Random Errors

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 $2^{m-r}$ 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 paper we give an efficient algorithm (in the block length $n=2^{m}$ ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree $r=o(\sqrt {m})$ ), 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 $r=o(\sqrt {m/\log m})$ ), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$ , our algorithm can correct any error pattern in $RM(m-(2r+2),m)$ , 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 Kotter that generalizes the Berlekamp–Welch algorithm for decoding Reed–Solomon codes.