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

04/20/2020

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by   Emmanuel Abbe, et al.

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This paper considers 'δ-almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a δ fraction of monomials of degree at most r. It is shown that for any δ > 0 and any ε>0, there exists a family of δ-almost Reed-Muller codes of constant rate that correct 1/2-ε 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 approach 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.