Reed–Muller Codes for Random Erasures and Errors

This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and in particular when can they achieve capacity 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. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on dierent techniques, which we nd interesting in their own right. In particular, we study the following questions about 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 random columns in E(m; r) that dene a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code C of suciently high rate we construct a new code C0 obtained by tensorizing C, such that for every subset S of coordinates, if C can recover from erasures in S, then C0 can recover from errors in S. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent [KLP12] bounds from constant degree to linear degree polynomials.