Date of Original Version

6-2011

Type

Conference Proceeding

Rights Management

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-22935-0_50

Abstract or Description

The classical family of [n,k] q Reed-Solomon codes over a field Fq consist of the evaluations of polynomials f∈Fq[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.

DOI

10.1007/978-3-642-22935-0_50

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Published In

Lecture Notes in Computer Science, 6845, 593-604.