Skip to main content
SpringerLink
Log in

Ideal Error-Correcting Codes: Unifying Algebraic and Number-Theoretic Algorithms

  • Conference paper
  • First Online:
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2227))

Abstract

Over the past five years a number of algorithms decoding some well-studied error-correcting codes far beyond their “error-correcting radii” have been developed. These algorithms, usually termed as listdecoding algorithms, originated with a list-decoder for Reed-Solomon codes [36],[17], and were soon extended to decoders for Algebraic Geometry codes [33],[17] and also to some number-theoretic codes [12],[6],[16]. In addition to their enhanced decoding capability, these algorithms enjoy the benefit of being conceptually simple, fairly general [16], and are capable of exploiting soft-decision information in algebraic decoding [24]. This article surveys these algorithms and highlights some of these features.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Sigal Ar, Richard J. Lipton, Ronitt Rubinfeld, and Madhu Sudan. Reconstructing algebraic functions from erroneous data. SIAM Journal on Computing, 28(2): 487–510, April 1999.

    Article  MathSciNet  Google Scholar 

  2. Daniel Augot and Lancelot Pecquet. A Hensel lifting to replace factorization inlist decoding of algebraic-geometric and Reed-Solomon codes. IEEE Trans. Info.Theory, 46(6): 2605–2613, November 2000.

    MATH  MathSciNet  Google Scholar 

  3. Elwyn R.Berlekamp. Bounded distance +1 soft-decision Reed Solomon decoding.IEEE Transactions on Information Theory, 42(3):704–720, 1996.

    Article  MATH  Google Scholar 

  4. Richard E. Blahut. Theory and practice of error control codes. Addison-Wesley Pub. Co., 1983.

    Google Scholar 

  5. V. M. Blinovskii. Bounds for codes in the case of list decoding of finite volume.Problemy Peradachi Informatsii, 22(1):11–25, January-March 1986.

    MathSciNet  Google Scholar 

  6. Dan Boneh. Finding smooth integers in short intervals using CRT decoding. (Toappear) Proceedings of the Thirty-Second Annual ACM Symposium on Theory ofComputing, Portland, Oregon, 21–23 May 2000.

    Google Scholar 

  7. Peter Elias. List decoding for noisy channels. WESCON Convention Record, Part2, Institute of Radio Engineers (now IEEE), pages 94–104, 1957.

    Google Scholar 

  8. Peter Elias. Error-correcting codes for list decoding. IEEE Transactions on InformationTheory, 37(1):5–12, January 1991.

    Article  MATH  MathSciNet  Google Scholar 

  9. G.-L. Feng and T. R. N. Rao. Decoding algebraic-geometric codes upto the designedminimum distance. IEEE Transactions on Information Theory, 39(1):37–45,January 1993.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. David Forney Jr.. Concatenated Codes. MIT Press, Cambridge, MA, 1966.

    Google Scholar 

  11. S. Gao and M. A. Shokrollahi. Computing roots of polynomials over function fields of curves. Proceedings of the Annapolis Conference on Number Theory, CodingTheory, and Cryptography, 1999.

    Google Scholar 

  12. Oded Goldreich, Dana Ron, and Madhu Sudan. Chinese remaindering with errors.IEEE Transactions on Information Theory. 46(4): 1330–1338, July 2000.

    Article  MATH  MathSciNet  Google Scholar 

  13. Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials withqueries: The highly noisy case. Proceedings of the 36th Annual Symposium onFoundations of Computer Science, pages 294–303, Milwaukee, Wisconsin, 23–25 October 1995.

    Google Scholar 

  14. Dima Grigoriev. Factorization of polynomials over a finite field and the solutionof systems of algebraic equations. Translated from Zapiski Nauchnykh SeminarovLenningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova ANSSSR, 137:20–79, 1984.

    Google Scholar 

  15. Venkatesan Guruswami, Johan Håstad, Madhu Sudan, and David Zuckerman.Combinatorial bounds for list decoding. (To appear) Proceedings of the 38th AnnualAllerton Conference on Communication, Control, and Computing, 2000.

    Google Scholar 

  16. Venkatesan Guruswami, Amit Sahai, and Madhu Sudan. “Soft-decision” decodingof Chinese remainder codes. Proceedings of the 41st Annual Symposium on Foundationsof Computer Science, pages 159–168, Redondo Beach, California, 12–14November, 2000.

    Google Scholar 

  17. Venkatesan Guruswami and Madhu Sudan. Improved decoding of Reed-Solomoncodes and algebraic-geometric codes. IEEE Transactions on Information Theory,45(6): 1757–1767, September 1999.

    Article  MATH  MathSciNet  Google Scholar 

  18. Venkatesan Guruswami and Madhu Sudan. List decoding algorithms for certainconcatenated codes. Proceedings of the Thirty-Second Annual ACM Symposiumon Theory of Computing, pages 181–190, Portland, Oregon, 21–23 May 2000.

    Google Scholar 

  19. Venkatesan Guruswami and Madhu Sudan, On representations of algebraicgeometriccodes. IEEE Transactions on Information Theory (to appear).

    Google Scholar 

  20. T. Høholdt, J. H. van Lint, and R. Pellikaan. Algebraic geometry codes. In Handbookof Coding Theory, V. Pless and C. Huffman (Eds.), Elsevier Sciences, 1998.

    Google Scholar 

  21. J. Justesen and T. Høholdt. Bounds on list decoding of MDS codes. Manuscript,1999.

    Google Scholar 

  22. Erich Kaltofen. A polynomial-time reduction from bivariate to univariate integralpolynomial factorization. Proceedings of the Fourteenth Annual ACM Symposiumon Theory of Computing, pages 261–266, San Francisco, California, 5–7 May 1982.

    Google Scholar 

  23. Ralf Kötter and Alexander Vardy. Algebraic soft-decision decoding of Reed-Solomon codes. (To appear) Proceedings of the 38th Annual Allerton Conferenceon Communication, Control, and Computing, 2000.

    Google Scholar 

  24. Arjen K. Lenstra. Factoring multivariate polynomials over finite fields. Journal ofComputer and System Sciences, 30(2):235–248, April 1985.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. K. Lenstra, H. W. Lenstra, and L. Lovasz. Factoring polynomials with rationalcoefficients. Mathematische Annalen, 261:515–534, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  26. D. M. Mandelbaum. On a class of arithmetic codes and a decoding algorithm.IEEE Transactions on Information Theory, 21:85–88, 1976.

    Article  MathSciNet  Google Scholar 

  27. R. Matsumoto. On the second step in the Guruswami-Sudan list decoding algorithmfor AG-codes. Technical Report of IEICE, pp. 65–70, 1999.

    Google Scholar 

  28. R. Refslund Nielsen and Tom Høholdt. Decoding Hermitian codes with Sudan’salgorithm. Proceedings of AAECC-13, LNCS 1719, Springer-Verlag, 1999, pp. 260–270.

    Google Scholar 

  29. W. W. Peterson. Encoding and error-correction procedures for Bose-Chaudhuricodes. IRE Transactions on Information Theory, IT-60:459–470, 1960.

    Article  Google Scholar 

  30. Ron M. Roth and Gitit Ruckenstein. Efficient decoding of Reed-Solomon codesbeyond half the minimum distance. IEEE Transactions on Information Theory,46(1):246–257, January 2000.

    Article  MATH  MathSciNet  Google Scholar 

  31. Madhu Sudan, Luca Trevisan, and Salil Vadhan. Pseudorandom generation withoutthe XOR lemma. Proceedings of the Thirty-First Annual ACM Symposiumon Theory of Computing, pages 537–546, Atlanta, Georgia, 1–4 May 1999.

    Google Scholar 

  32. M. Amin Shokrollahi and Hal Wasserman. List decoding of algebraic-geometriccodes. IEEE Transactions on Information Theory, 45(2): 432–437, March 1999.

    Article  MATH  MathSciNet  Google Scholar 

  33. Henning Stichtenoth. Algebraic Function Fields and Codes. Springer-Verlag, Berlin,1993.

    MATH  Google Scholar 

  34. Madhu Sudan. Efficient Checking of Polynomials and Proofs and the Hardness ofApproximations. ACM Distinguished Theses. Lecture Notes in Computer Science,no. 1001, Springer, 1996.

    Google Scholar 

  35. Madhu Sudan. Decoding of Reed Solomon codes beyond the error-correctionbound. Journal of Complexity, 13(1): 180–193, March 1997.

    Article  MATH  MathSciNet  Google Scholar 

  36. Lloyd Welch and Elwyn R. Berlekamp. Error correction of algebraic block codes. US Patent Number 4,633,470, issued December 1986.

    Google Scholar 

  37. J. M. Wozencraft. List decoding. Quarterly Progress Report. Research Laboratory of Electronics, MIT, Vol. 48, pp. 90–95, 1958.

    Google Scholar 

  38. Xin-Wen Wu and Paul H. Siegel. Efficient list decoding of algebraic geometriccodes beyond the error correction bound. Proc. of International Symposium onInformation Theory, June 2000.

    Google Scholar 

  39. V. V. Zyablov and M. S. Pinsker. List cascade decoding. Problemy Peredachi Informatsii,17(4):29–33, October-December 1981.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MIT Laboratory for Computer Science, 200 Technology Square, 02139, Cambridge, MA, USA

    Madhu Sudan

Authors
  1. Madhu Sudan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, RMIT University, GPO Box 2476V, 3001, Melbourne, Australia

    Serdar Boztaş

  2. Department of Computing, Macquarie University, 2109, NSW, Australia

    Igor E. Shparlinski

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sudan, M. (2001). Ideal Error-Correcting Codes: Unifying Algebraic and Number-Theoretic Algorithms. In: Boztaş, S., Shparlinski, I.E. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2001. Lecture Notes in Computer Science, vol 2227. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45624-4_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-45624-4_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42911-1

  • Online ISBN: 978-3-540-45624-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation