Almost-Reed-Muller Codes Achieve Constant Rates for Random Errors

This paper considers ' \delta -Almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a \delta fraction of monomials of degree at most d. It is shown that for any \delta > 0 and any \varepsilon >0 , there exists a family of \delta -Almost Reed-Muller codes of constant rate that correct 1/2-\varepsilon fraction of random errors with high probability. For exact Reed-Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed-Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC '15). Our proof is based on the recent polarization result for Reed-Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed-Muller code entropies.