List-decodability with large radius for Reed-Solomon codes

TL;DR

This paper demonstrates that Reed-Solomon codes can be constructed with optimal trade-offs between rate and list-decoding radius near the maximum, using polynomially large fields, improving previous results.

Contribution

It establishes the existence of Reed-Solomon codes with optimal rate-radius trade-offs for list-decoding close to the maximum radius, requiring only polynomial field size.

Findings

Reed-Solomon codes with rate proportional to epsilon are list-decodable up to radius 1-epsilon.

Achieves the optimal trade-off between rate and list size for large-radius decoding.

Reduces field size requirements from exponential to polynomial in block length.

Abstract

List-decodability of Reed-Solomon codes has received a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form $r=1-\varepsilon$ for $\varepsilon$ tending to zero. Our main result states that there exist Reed-Solomon codes with rate $\Omega(\varepsilon)$ which are $(1-\varepsilon, O(1/\varepsilon))$-list-decodable, meaning that any Hamming ball of radius $1-\varepsilon$ contains at most $O(1/\varepsilon)$ codewords. This trade-off between rate and list-decoding radius is best-possible for any code with list size less than exponential in the block length. By achieving this trade-off between rate and list-decoding radius we improve a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main motivating question of their work. Moreover, while their result requires the field to be exponentially large in the block length, we only need the field size to be polynomially large (and in fact, almost-linear suffices). We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of any given code with very large distance.