Optimal Rate List Decoding via Derivative Codes

  • Venkatesan Guruswami
  • Carol Wang
Conference paper

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

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6845)
Cite this paper as:
Guruswami V., Wang C. (2011) Optimal Rate List Decoding via Derivative Codes. In: Goldberg L.A., Jansen K., Ravi R., Rolim J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

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

Personalised recommendations