Year: 2020
Type: preprint
Abstract: 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 $d$. It is shown that for any $δ> 0$ and any $\varepsilon>0$, there exists a family of $δ$-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.
Source: arXiv (Cornell University)
Authors: Abbe, Emmanuel, Hązła, Jan, Nachum, Ido
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3020744928