Weighted Reed–Muller codes revisited (original) (raw)

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References

1. Andersen H.E., Geil O.: Evaluation codes from order domain theory. Finite Fields Th. App. 14, 92–123 (2008)
   Article MathSciNet MATH Google Scholar
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).
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)
   Chapter Google Scholar
4. Beelen P., Brander K.: Efficient list decoding of a class of algebraic-geometry codes. Adv. Math. Commun. 4, 485–518 (2010)
   Article MathSciNet MATH Google Scholar
5. DeMillo R.A., Lipton R.J.: A probabilistic remark on algebraic program testing. Inf. Process. Lett. 7(4), 193–195 (1978)
   Article MATH Google Scholar
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).
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)
   Article MathSciNet MATH Google Scholar
8. Feng G.-L., Rao T.R.N.: Improved geometric Goppa codes, Part I: basic theory. IEEE Trans. Inf. Theory 41, 1678–1693 (1995)
   Article MathSciNet MATH Google Scholar
9. Geil O., Høholdt, T.: On hyperbolic codes. Proc. AAECC-14, Lecture Notes in Comput. Sci. 2227, 159–171 (2001).
   Google Scholar
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).
11. Geil O., Thomsen C.: Tables for numbers of zeros with multiplicity at least r, webpage: http://zeros.spag.dk, January 18th (2011).
12. Guruswami V., Sudan M.: Improved decoding of Reed–Solomon and algebraic-geometry codes. IEEE Trans. Inf. Theory 45, 1757–1767 (1999)
    Article MathSciNet MATH Google Scholar
13. Hansen J.P.: Toric varieties Hirzebruch surfaces and error-correcting codes. Appl. Algebra Eng. Comm. Comput. 13, 289–300 (2002)
    Article MATH Google Scholar
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. Joyner D.: Toric codes over finite fields. Appl. Algebra Eng. Comm. Comput. 15, 63–79 (2004)
    Article MathSciNet MATH Google Scholar
16. Kabatiansky G.: Two Generalizations of prcduct Codes. Proc. of Acad. Sci. USSR, Cybern. Theory Regul. 232, vol. 6, pp. 1277–1280 (1977).
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)
    Article MathSciNet MATH Google Scholar
18. Little J., Schenck H.: Toric surface codes and Minkowski sums. SIAM J. Discret. Math. 20, 999–1014 (2007)
    Article MathSciNet Google Scholar
19. Lidl R., Niederreiter H.: Introduction to finite fields and their applications. University of Cambridge Press, New York (1986)
    MATH Google Scholar
20. Massey J., Costello D.J., Justesen J.: Polynomial weights and code constructions. IEEE Trans. Inf. Theory 19, 101–110 (1973)
    Article MathSciNet MATH Google Scholar
21. Pellikaan R., Wu X.-W.: List decoding of _q_-ary Reed–Muller codes. IEEE Trans. Inf. Theory 50, 679–682 (2004)
    Article MathSciNet Google Scholar
22. Pellikaan R., Wu X.-W.: List decoding of _q_-ary Reed–Muller codes. (Expanded version of the paper [21]), available from http://win.tue.nl/~ruudp/paper/43-exp.pdf, p. 37 (2004).
23. Ruano D.: On the parameters of _r_-dimensional toric codes. Finite Fields and their Applications 13, 962–976 (2007)
    Article MathSciNet MATH Google Scholar
24. Ruano D.: On the structure of generalized toric codes. J. Symb. Comput. 44, 499–506 (2009)
    Article MathSciNet MATH Google Scholar
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).
26. Schwartz J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach. 27(4), 701–717 (1980)
    Article MathSciNet MATH Google Scholar
27. Sørensen A.B.: Weighted Reed–Muller codes and algebraic-geometric codes. IEEE Trans. Inf. Theory 38, 1821–1826 (1992)
    Article Google Scholar
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).
29. Zippel, R.: Probabilistic algorithms for sparse polynomials. Proc. of EUROSAM 1979, Lecture Notes in Comput. Sci., 72. Springer, Berlin, p. 216–226 (1979).

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