Optimal rate list decoding via derivative codes

TL;DR

This paper introduces derivative codes, a new family related to Reed-Solomon codes, and demonstrates their ability to be list-decoded efficiently near the optimal error correction radius, offering an alternative to folded Reed-Solomon codes.

Contribution

The paper presents derivative codes and a polynomial-time list decoding algorithm that achieves near-optimal error correction, providing a new approach to high-rate code decoding.

Findings

Codes can be list-decoded from errors approaching the rate R

Decoding algorithm is linear-algebraic and efficient

Offers advantages over folded Reed-Solomon codes in certain decoding scenarios

Abstract

The classical family of $[n,k]_q$ Reed-Solomon codes over a field $\F_q$ consist of the evaluations of polynomials $f \in \F_q[X]$ of degree $< k$ at $n$ distinct field elements. In this work, we consider a closely related family of codes, called (order $m$) {\em derivative codes} and defined over fields of large characteristic, which consist of the evaluations of $f$ as well as its first $m-1$ formal derivatives at $n$ distinct field elements. For large enough $m$, we show that these codes can be list-decoded in polynomial time from an error fraction approaching $1-R$, where $R=k/(nm)$ is the rate of the code. This gives an alternate construction to folded Reed-Solomon codes for achieving the optimal trade-off between rate and list error-correction radius. Our decoding algorithm is linear-algebraic, and involves solving a linear system to interpolate a multivariate polynomial, and then solving another structured linear system to retrieve the list of candidate polynomials $f$. The algorithm for derivative codes offers some advantages compared to a similar one for folded Reed-Solomon codes in terms of efficient unique decoding in the presence of side information.