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

This paper considers “δ-almost Reed–Muller codes”, i.e., linear codes spanned by evaluations of all but aδfraction of monomials of degree at mostd. It is shown that for anyδ > 0and anyε >0, there exists a family ofδ-almost Reed–Muller codes of constant rate that correct1/2- εfraction 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.