Designs, Codes and Cryptography

, Volume 66, Issue 1–3, pp 195–220 | Cite as

Weighted Reed–Muller codes revisited

  • Olav Geil
  • Casper Thomsen


We consider weighted Reed–Muller codes over point ensemble S 1 × · · · × S m where S i needs not be of the same size as S j . For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S 1|/|S 2| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.


Affine variety codes List decoding Weighted Reed–Muller codes 

Mathematics Subject Classification

11T71 94B35 12E20 


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  1. 1.
    Andersen H.E., Geil O.: Evaluation codes from order domain theory. Finite Fields Th. App. 14, 92–123 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Augot D., El-Khamy M., McEliece R.J., Parvaresh F., Stepanov M., Vardy A.: List decoding of Reed– Solomon product codes. In: Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory, pp. 210–213. Zvenigorod (2006).Google Scholar
  3. 3.
    Augot D., Stepanov M.: A note on the generalisation of the Guruswami–Sudan list decoding algorithm to Reed–Muller codes. In: Mora, L., Sala, M., Sakata, S., Perret, L., Traverso, C. (eds.) Gröbner Bases, Coding, and Cryptography, pp. 395–398. Springer, Berlin (2009)CrossRefGoogle Scholar
  4. 4.
    Beelen P., Brander K.: Efficient list decoding of a class of algebraic-geometry codes. Adv. Math. Commun. 4, 485–518 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    DeMillo R.A., Lipton R.J.: A probabilistic remark on algebraic program testing. Inf. Process. Lett. 7(4), 193–195 (1978)zbMATHCrossRefGoogle Scholar
  6. 6.
    Dvir Z., Kopparty S., Saraf S., Sudan M.: Extensions to the method of multiplicities, with applications to Kakeya sets and Mergers, (appeared in Proc. of FOCS 2009) arXiv:0901.2529v2, p. 26 (2009).Google Scholar
  7. 7.
    Feng G.-L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inf. Theory 40, 1003–1012 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Feng G.-L., Rao T.R.N.: Improved geometric Goppa codes, Part I: basic theory. IEEE Trans. Inf. Theory 41, 1678–1693 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Geil O., Høholdt, T.: On hyperbolic codes. Proc. AAECC-14, Lecture Notes in Comput. Sci. 2227, 159–171 (2001).Google Scholar
  10. 10.
    Geil O., Matsumoto R.: Generalized Sudan’s list decoding for order domain codes. Proc. AAECC-16, Lecture Notes in Comput. Sci., 4851, pp. 50–59. Springer, Berlin (2007).Google Scholar
  11. 11.
    Geil O., Thomsen C.: Tables for numbers of zeros with multiplicity at least r, webpage:, January 18th (2011).
  12. 12.
    Guruswami V., Sudan M.: Improved decoding of Reed–Solomon and algebraic-geometry codes. IEEE Trans. Inf. Theory 45, 1757–1767 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hansen J.P.: Toric varieties Hirzebruch surfaces and error-correcting codes. Appl. Algebra Eng. Comm. Comput. 13, 289–300 (2002)zbMATHCrossRefGoogle Scholar
  14. 14.
    Høholdt T., van Lint J., Pellikaan R.: Algebraic geometry codes, Chap. 10. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory. vol. 1, pp. 871–961. Elsevier, Amsterdam (1998)Google Scholar
  15. 15.
    Joyner D.: Toric codes over finite fields. Appl. Algebra Eng. Comm. Comput. 15, 63–79 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kabatiansky G.: Two Generalizations of prcduct Codes. Proc. of Acad. Sci. USSR, Cybern. Theory Regul. 232, vol. 6, pp. 1277–1280 (1977).Google Scholar
  17. 17.
    Kasami T., Lin S., Peterson W.: New generalizations of the Reed–Muller codes. I. Primitive codes. IEEE Trans. Inf. Theory 14, 189–199 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Little J., Schenck H.: Toric surface codes and Minkowski sums. SIAM J. Discret. Math. 20, 999–1014 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lidl R., Niederreiter H.: Introduction to finite fields and their applications. University of Cambridge Press, New York (1986)zbMATHGoogle Scholar
  20. 20.
    Massey J., Costello D.J., Justesen J.: Polynomial weights and code constructions. IEEE Trans. Inf. Theory 19, 101–110 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pellikaan R., Wu X.-W.: List decoding of q-ary Reed–Muller codes. IEEE Trans. Inf. Theory 50, 679–682 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pellikaan R., Wu X.-W.: List decoding of q-ary Reed–Muller codes. (Expanded version of the paper [21]), available from, p. 37 (2004).
  23. 23.
    Ruano D.: On the parameters of r-dimensional toric codes. Finite Fields and their Applications 13, 962–976 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ruano D.: On the structure of generalized toric codes. J. Symb. Comput. 44, 499–506 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Santhi N.: On algebraic decoding of q-ary Reed–Muller and product-Reed–Solomon codes. In: Proc. IEEE Int. Symp. Inf. Th., Nice, pp. 1351–1355 (2007).Google Scholar
  26. 26.
    Schwartz J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach. 27(4), 701–717 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sørensen A.B.: Weighted Reed–Muller codes and algebraic-geometric codes. IEEE Trans. Inf. Theory 38, 1821–1826 (1992)CrossRefGoogle Scholar
  28. 28.
    Wu X.-W.: An algorithm for finding the roots of the polynomials over order domains. In: Proc. of 2002, IEEE Int. Symp. Inf. Th., Lausanne (2002).Google Scholar
  29. 29.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. Proc. of EUROSAM 1979, Lecture Notes in Comput. Sci., 72. Springer, Berlin, p. 216–226 (1979).Google Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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