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

Abstract

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 $\epsilon >0$, there exists a family of $\delta$-almost Reed-Muller codes of constant rate that correct $1/2-\epsilon$ 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.

References in corpus (1)

• Comparison of Channels: Criteria for Domination by a Symmetric Channel

Cited by in corpus (1)

• On codes decoding a constant fraction of errors on the BSC