Abstract
The complexity of decoding the standard Reed-Solomon code is a well known open problem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance).
Similar content being viewed by others
References
Sudan M. Decoding of Reed-Solomon codes beyond the error-correction bound. J Complexity, 13: 180–193 (1997)
Guruswami V, Sudan M. Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Trans Inform Theory, 45(6): 1757–1767 (1999)
Guruswami V, Vardy A. Maximum-likelihood decoding of Reed-Solomon codes is NP-hard. IEEE Trans Inform Theory, 51(7): 2249–2256 (2005)
Li J, Wan D. On the subset sum problem over finite fields. Finite Fields Appl, to appear
Cheng Q, Murray E. On deciding deep holes of Reed-Solomon codes. In: Proceedings of TAMC 2007, LNCS 4484. Berlin: Springer, 296–305
Cheng Q, Wan D. On the list and bounded distance decodibility of the Reed-Solomon codes (extended abstract). In: Proc 45th IEEE Symp on Foundation of Comp Sciences (FOCS). Washington: IEEE Society, 2004, 335–341
Wan D. Generators and irreducible polynomials over finite fields. Math Comp, 66(219): 1195–1212 (1997)
Cheng Q, Wan D. On the list and bounded distance decodibility of Reed-Solomon codes. SIAM J Comput, 37(1): 195–209 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, Y., Wan, D. On error distance of Reed-Solomon codes. Sci. China Ser. A-Math. 51, 1982–1988 (2008). https://doi.org/10.1007/s11425-008-0066-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-008-0066-3