Efficiently list-decodable punctured Reed-Muller codes

TL;DR

This paper presents explicit, rate-efficient puncturings of Reed-Muller codes that nearly preserve their distance and list decodability, achieving improved parameters over prior constructions, especially for small degree relative to the field size.

Contribution

It introduces explicit puncturing methods for Reed-Muller codes that maintain strong list decoding properties with better rate and field size conditions, using algebraic geometry and concatenation techniques.

Findings

Achieves rate (/!) with near-optimal field size

Maintains relative distance at least (1-)

Enables list decoding up to (1-) error fraction

Abstract

The Reed-Muller (RM) code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ for $d < q$, with its evaluation on ${\mathbb F}_q^n$, has relative distance $1-d/q$ and can be list decoded from a $1-O(\sqrt{d/q})$ fraction of errors. In this work, for $d \ll q$, we give a length-efficient puncturing of such codes which (almost) retains the distance and list decodability properties of the Reed-Muller code, but has much better rate. Specificially, when $q =\Omega( d^2/\epsilon^2)$, we given an explicit rate $\Omega\left(\frac{\epsilon}{d!}\right)$ puncturing of Reed-Muller codes which have relative distance at least $(1-\epsilon)$ and efficient list decoding up to $(1-\sqrt{\epsilon})$ error fraction. This almost matches the performance of random puncturings which work with the weaker field size requirement $q= \Omega( d/\epsilon^2)$. We can also improve the field size requirement to the optimal (up to constant factors) $q =\Omega( d/\epsilon)$, at the expense of a worse list decoding radius of $1-\epsilon^{1/3}$ and rate $\Omega\left(\frac{\epsilon^2}{d!}\right)$. The first of the above trade-offs is obtained by substituting for the variables functions with carefully chosen pole orders from an algebraic function field; this leads to a puncturing for which the RM code is a subcode of a certain algebraic-geometric code (which is known to be efficiently list decodable). The second trade-off is obtained by concatenating this construction with a Reed-Solomon based multiplication friendly pair, and using the list recovery property of algebraic-geometric codes.