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On Deciding Deep Holes of Reed-Solomon Codes

  • Qi Cheng
  • Elizabeth Murray
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4484)

Abstract

For generalized Reed-Solomon codes, it has been proved [7] that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code – a property that practical codes do not usually possess. In this paper, we first present a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector (f(α))_α ∈ F p for the Reed-Solomon [p − 1,k] p , k < p 1/4 − ε , cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d < p 3/13 − ε .

Keywords

Reed-Solomon codes deep hole NP-complete algebraic surface 

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References

  1. 1.
    Berlekamp, E., Welch, L.: Error correction of algebraic block codes. U.S. Patent Number 4633470 (1986)Google Scholar
  2. 2.
    Cafure, A., Matera, G.: Improved explicit estimates on the number of solutions of equations over a finite field (2004), http://www.arxiv.org/abs/math.NT/0405302
  3. 3.
    Cheng, Q., Wan, D.: On the list and bounded distance decodibility of the Reed-Solomon codes (extended abstract). In: Proc. 45th IEEE Symp. on Foundations of Comp. Science, FOCS, pp. 335–341 (2004)Google Scholar
  4. 4.
    Cohen, S.D.: Explicit theorems on generator polynomials. Finite Fields and Their Applications. To appear (2005)Google Scholar
  5. 5.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Guruswami, V., Sudan, M.: Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Transactions on Information Theory 45(6), 1757–1767 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Guruswami, V., Vardy, A.: Maximum-likelihood decoding of Reed-Solomon codes is NP-hard. In: Proceeding of SODA (2005)Google Scholar
  8. 8.
    Schmidt, W.: Equations over Finite Fields. An Elementary Approach. Lecture Notes in Mathematics, vol. 536. Springer, Heidelberg (1976)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Qi Cheng
    • 1
  • Elizabeth Murray
    • 1
  1. 1.School of Computer Science, The University of Oklahoma, Norman, OK 73019USA

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