Efficient list decoding of punctured Reed-Muller codes

The Reed-Muller (RM) code encoding n-variate degree-d polynomials over Fq for d < q, with its evaluation on Fq , has relative distance 1− d/q and can be list decoded from a 1−O( √ d/q) fraction of errors. In this work, for d ≪ q, we give a length-efficient puncturing of such codes which (almost) retains the distance and list decodability properties of the Reed-Muller code, but has much better rate. Specificially, when q = Ω(d/ε), we given an explicit rate Ω ( ε d! ) puncturing of Reed-Muller codes which have relative distance at least (1 − ε) and efficient list decoding up to (1 − √ε) error fraction. This almost matches the performance of random puncturings which work with the weaker field size requirement q = Ω(d/ε). We can also improve the field size requirement to the optimal (up to constant factors) q = Ω(d/ε), at the expense of a worse list decoding radius of 1− ε and rate Ω (