Reed-Muller Codes for Random Erasures and Errors
Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15
This paper studies the parameters for which Reed-Muller codes over GF (2) can correct random erasures and random errors with high probability, and in particular when can they achieve capacity (namely, the information theoretic limit) for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF (2) polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes.
... errors, we prove that RM codes achieve capacity for very low rate. For very high rate we prove a weaker result: RM codes uniquely decode far beyond the minimum distance, from about square root of the number of random errors possible at capacity. The (very different) proofs of these four results require in particular the following technical results, which we find interesting in their own right. For erasures, we study the following two natural problems concerning E(m, r), the matrix whose rows are truth tables of all monomials of degree ≤ r in m variables. What is the most (resp. least) number of columns of E(m, r), that if chosen at random, the submatrix they define will have, with high probability, full column rank (resp. full row rank)? We obtain tight bounds for very small (resp. very large) degrees r, which imply that RM codes achieve capacity for erasures for very high (resp. very low) rates. Our decoding from random errors follows from the following novel reduction. For every linear code C of sufficiently high rate we construct a new code C , also of very high rate, such that for every subset S of coordinates, if C can recover from erasures of codewords in coordinates S, then C can recover from errors in coordinates S. Specializing (and improving the parameters of) this reduction to Reed-Muller codes, combined with our results on erasures, implies our result on unique decoding of RM codes of high rate from random errors. Two of our capacity achieving results require tight bounds on the weight distribution of Reed-Muller codes. We obtain such bounds extending the recent [KLP12] bounds from constant degree to linear degree polynomials.