Optimal Rate List Decoding via Derivative Codes

Abstract

The classical family of [n,k] _q Reed-Solomon codes over a field $\mathbb{F}_q$ consist of the evaluations of polynomials $f \in \mathbb{F}_q[X]$ of degree < k at n distinct field elements. In this work, we consider a closely related family of codes, called (order m) 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.

Research supported in part by NSF CCF-0963975 and MSR-CMU Center for Computational Thinking.