Approximate local decoding of cubic reed-muller codes beyond the list decoding radius

Abstract

We consider the question of decoding Reed-Muller codes over Fn2 beyond their list-decoding radius. Since, by definition, in this regime one cannot demand an efficient exact listdecoder, we seek an approximate decoder: Given a word F and radii r0 > r > 0, the goal is to output a codeword within radius r0 of F, if there exists a codeword within distance r. As opposed to the list decoding problem, it suffices here to output any codeword with this property, since the list may be too large if r exceeds the list decoding radius. Prior to our work, such decoders were known for ReedMuller codes of degree 2, due to works of Wolf and the second author [FOCS 2011]. In this work we make the first progress on this problem for the degree 3 where the list decoding radius is 1=8. We show that there is a constant = 1=2 p 1=8 > 1=8 and an efficient approximate decoder, that given query access to a function F : Fn2 ! F2, such that F is within distance r =ϵ from a cubic polynomial, runs in time polynomial in message length and outputs with high probability a cubic polynomial which is at distance at most r0 = 1=2-ϵ0 from F, where "0 is a quasi polynomial function of ϵ.