Optimal Rate List Decoding via Derivative Codes

  • Venkatesan Guruswami
  • Carol Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6845)

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.

Keywords

Reed-Solomon codes list error-correction noisy polynomial interpolation multiplicity codes subspace-evasive sets pseudorandomness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Carol Wang
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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