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.


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


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  1. 1.
    Bombieri, E., Kopparty, S.: List decoding multiplicity codes (2011) (manuscript)Google Scholar
  2. 2.
    Gemmell, P., Sudan, M.: Highly resilient correctors for multivariate polynomials. Information Processing Letters 43(4), 169–174 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Guruswami, V.: List decoding with side information. In: Proceedings of the 18th IEEE Conference on Computational Complexity (CCC), pp. 300–309 (2003)Google Scholar
  4. 4.
    Guruswami, V.: Algorithmic Results in List Decoding. Foundations and Trends in Theoretical Computer Science (FnT-TCS), vol. 2. NOW publishers (January 2007)Google Scholar
  5. 5.
    Guruswami, V.: Cyclotomic function fields, Artin-Frobenius automorphisms, and list error-correction with optimal rate. Algebra and Number Theory 4(4), 433–463 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guruswami, V.: Linear-algebraic list decoding of folded Reed-Solomon codes. In: Proceedings of the 26th IEEE Conference on Computational Complexity (June 2011)Google Scholar
  7. 7.
    Guruswami, V., Rudra, A.: Limits to list decoding Reed-Solomon codes. IEEE Transactions on Information Theory 52(8), 3642–3649 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Guruswami, V., Rudra, A.: Explicit codes achieving list decoding capacity: Error-correction up to the Singleton bound. IEEE Transactions on Information Theory 54(1), 135–150 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guruswami, V., Sudan, M.: Improved decoding of Reed-Solomon and Algebraic-geometric codes. IEEE Transactions on Information Theory 45(6), 1757–1767 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kopparty, S., Saraf, S., Yekhanin, S.: High-rate codes with sublinear-time decoding. Electronic Colloquium on Computational Complexity, TR10-148 (2010)Google Scholar
  11. 11.
    Parvaresh, F., Vardy, A.: Correcting errors beyond the Guruswami-Sudan radius in polynomial time. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 285–294 (2005)Google Scholar
  12. 12.
    Sudan, M.: Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity 13(1), 180–193 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vadhan, S.: Pseudorandomness. Foundations and Trends in Theoretical Computer Science (FnT-TCS). NOW publishers (2010) (to appear) Draft available at,
  14. 14.
    Welch, L.R., Berlekamp, E.R.: Error correction of algebraic block codes. US Patent Number 4, 633, 470 (December 1986)Google Scholar

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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