# Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

We prove the existence of Reed-Solomon codes of any desired rate R ∈ (0,1) that are combinatorially list-decodable up to a radius approaching 1-R, which is the information-theoretic limit. This is established by starting with the full-length [q,k]_q Reed-Solomon code over a field 𝔽_q that is polynomially larger than the desired dimension k, and "puncturing" it by including k/R randomly chosen codeword positions. Our puncturing result is more general and applies to any code with large minimum distance: we show that a random rate R puncturing of an 𝔽_q-linear "mother" code whose relative distance is close enough to 1-1/q is list-decodable up to a radius approaching the q-ary list-decoding capacity bound h_q^-1(1-R). In fact, for large q, or under a stronger assumption of low-bias of the mother-code, we prove that the threshold rate for list-decodability with a specific list-size (and more generally, any "local" property) of the random puncturing approaches that of fully random linear codes. Thus, all current (and future) list-decodability bounds shown for random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) code. This can be viewed as a general derandomization result applicable to random linear codes. To obtain our conclusion about Reed-Solomon codes, we establish some hashing properties of field trace maps that allow us to reduce the list-decodability of RS codes to its associated trace (dual-BCH) code, and then apply our puncturing theorem to the latter. Our approach implies, essentially for free, optimal rate list-recoverability of punctured RS codes as well.

• 49 publications
• 9 publications
05/31/2021

### List-decoding and list-recovery of Reed-Solomon codes beyond the Johnson radius for any rate

Understanding the limits of list-decoding and list-recovery of Reed-Solo...
05/04/2020

### On the list recoverability of randomly punctured codes

We show that a random puncturing of a code with good distance is list re...
12/19/2020

### List-decodability with large radius for Reed-Solomon codes

List-decodability of Reed-Solomon codes has received a lot of attention ...
11/09/2020

### Improved List-Decodability of Reed–Solomon Codes via Tree Packings

This paper shows that there exist Reed–Solomon (RS) codes, over large fi...
09/09/2020

### Sharp threshold rates for random codes

Suppose that 𝒫 is a property that may be satisfied by a random code C ⊂Σ...
02/20/2020

### Linear-time Erasure List-decoding of Expander Codes

We give a linear-time erasure list-decoding algorithm for expander codes...
07/02/2019

### On the list decodability of Rank Metric codes

Let $k,n,m \in \mathbb{Z}^+$ integers such that $k\leq n \leq m$, let \$\...