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arXiv · 2004.09590

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

Emmanuel Abbe·Jan Hązła·Ido Nachum
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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 $\varepsilon>0$, 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 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.

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Emmanuel Abbe, Jan Hązła, Ido Nachum. 2020-04-20. Almost-Reed--Muller Codes Achieve Constant Rates for Random Errors. https://arxiv.org/abs/2004.09590

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