Almost-Reed--Muller Codes Achieve Constant Rates for Random Errors
Information Theory
2021-10-07 v2 math.IT
Abstract
This paper considers '-almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a fraction of monomials of degree at most . It is shown that for any and any , there exists a family of -almost Reed-Muller codes of constant rate that correct 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.
Cite
@article{arxiv.2004.09590,
title = {Almost-Reed--Muller Codes Achieve Constant Rates for Random Errors},
author = {Emmanuel Abbe and Jan Hązła and Ido Nachum},
journal= {arXiv preprint arXiv:2004.09590},
year = {2021}
}