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An improved list decoding algorithm for the second order Reed–Muller codes and its applications

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Abstract

We propose an algorithm which is an improved version of the Kabatiansky–Tavernier list decoding algorithm for the second order Reed–Muller code RM(2, m), of length n = 2m, and we analyse its theoretical and practical complexity. This improvement allows a better theoretical complexity. Moreover, we conjecture another complexity which corresponds to the results of our simulations. This algorithm has the strong property of being deterministic and this fact drives us to consider some applications, like determining some lower bounds concerning the covering radius of the RM(2, m) code.

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References

  1. Elias P. (1957). List decoding for noisy channels. 1957-IRE WESCON Conv. Record 2, 94–104

    Google Scholar 

  2. Guruswami V., Sudan M. (1999). Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Trans. Inform. Theory 45, 1757–1767

    Article  MATH  MathSciNet  Google Scholar 

  3. Goldreich O., Levin L.A.: A hard-core predicate for all one-way functions. In: Proceedings of 21st ACM Symp. on Theory of Computing, pp. 25–32 (1989).

  4. Goldreich O., Rubinfeld R., Sudan M.: Learning polynomials with queries: the highly noisy case. In: Proceedings of 36th Symp. on Foundations of Computer Science, pp. 294–303 (1995).

  5. Pellikaan R., Wu X.-W. (2004). List decoding of q-ary Reed–Muller Codes. IEEE Trans. Inform. Theory 50, 679–682

    Article  MathSciNet  Google Scholar 

  6. Kabatiansky G., Tavernier C.: List decoding of second order Reed–Muller Codes. In: Proceedings of the 8th Intern. Simp. Comm. Theory and Applications, Ambleside, UK, July 2005.

  7. Dumer I., Kabatiansky G., Tavernier C.: List decoding of Reed–Muller codes up to the Johnson bound with almost linear complexity. In: Proceedings of the ISIT 2006. Seattle, USA (2006).

  8. Carlet C., Mesnager S. (2007). Improving the upper bounds on the covering radii of binary Reed–Muller codes. IEEE Trans. Inform. Theory 53, 162–173

    Article  MathSciNet  Google Scholar 

  9. Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes. North-Holland (1997).

  10. Carlet C.: The complexity of Boolean functions from cryptographic viewpoint. In: Krause M., Pudlak P., Reischuk R., van Melkebeek D. (eds.) Dagstuhl Seminar “Complexity of Boolean Functions”, 15 pp. (2006).

  11. Dumer I. (2004). Recursive decoding and its performance for low-rate Reed–Muller codes. IEEE Trans. Inform. Theory 50, 811–823

    Article  MathSciNet  Google Scholar 

  12. Sloane N., Berlekamp E. (1970). Weight enumerator for second-order Reed–Muller codes. IEEE Trans. Inform. Theory 16(6): 745–751

    Article  MATH  MathSciNet  Google Scholar 

  13. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company (1977).

  14. Kabatiansky G., Tavernier C.: List decoding of first order Reed–Muller codes II. In: Proceedings of the ACCT-10. Zvenigorod, Russia (2006).

  15. Hou X.D. (1993). Some results on the covering radii of Reed–Muller codes. IEEE Trans. Inform. Theory 39(2): 366–378

    Article  MATH  MathSciNet  Google Scholar 

  16. Schatz J. (1981). The second-order Reed–Muller code of length 64 has covering radius 18. IEEE Trans. Inform. Theory IT-27(4): 529530

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris 8 Saint-Denis (MAATICAH), 2, rue de la Liberté, Saint-Denis cedex, 93526, France

    Rafaël Fourquet

  2. THALES Communications, Colombes, France

    Cédric Tavernier

Authors
  1. Rafaël Fourquet
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  2. Cédric Tavernier
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Correspondence to Rafaël Fourquet.

Additional information

This article is a rewritten and completed version of Fourquet R., Tavernier C.: List decoding of second order Reed–Muller codes and its covering radius implications, Workshop on Coding and Cryptography 2007, pp. 147–156.

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Fourquet, R., Tavernier, C. An improved list decoding algorithm for the second order Reed–Muller codes and its applications. Des. Codes Cryptogr. 49, 323–340 (2008). https://doi.org/10.1007/s10623-008-9184-8

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  • DOI: https://doi.org/10.1007/s10623-008-9184-8

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