Abstract
To obtain upper bounds on the distance of a binary linear code, many probabilistic algorithms have been proposed. The author presents a general variation to these algorithms, specific for cyclic codes, which is shown to be an improvement. As an example, the author optimizes Brouwer's algorithm to find the best upper bounds on the dual distance of BCH[255,k,d].
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References
D. Augot, Newton's identities for minimum codewords of a family of alternant codes, Short Abstract in the Proceedings of IEEE ISIT 95 (1995).
D. Augot and F. Levy-dit-Vehel, Bounds on the minimum distance of the duals of BCH codes, IEEE Trans. on Inf. Th., Vol. 42, No. 4, July (1996) pp. 1257–1260.
E. R. Berlekamp, Algebraic Coding Theory, New York, McGraw-Hill (1968).
A. M. Barg and I. I. Dumer, On computing the weight spectrum of cyclic codes, IEEE Trans. on Inf. Th., Vol. 38, No. 4, July (1992) pp. 1382–1386.
M. Becker and J. Cramwinckel, Implementation of an algorithm for the weight distribution of block codes, Modelleringcolloquium, Eindhoven Univ. Technology (1995).
A. E. Brower and T. Verhoeff, An updated table of minimum distance bounds for binary linear codes, IEEE. Trans. on Inf. Th., Vol. 39, No. 2, March (1993) pp. 662–677.
A. Canteaut and F. Chabaud. A new algorithm for finding minimum-weight words in a linear code: application to McEliece's cryptosystemand to narrow-sense BCH codes of length 511, IEEE Trans. on Inf. Th., Vol. 44, No. 1, January (1998) pp. 367–378.
L. Carlitz and S. Uchiyama, Bounds for exponential sums, Duke Math. J., Vol. 24, December (1957) pp. 37–41.
Y. Desaki, T. Fujiwara and T. Kasami, The weight distributions of extended binary primitive BCH codes of length 128, IEEE Trans. on Inf. Th., Vol. 43, No. 4, July (1997) pp. 1364–1371.
T. Fujiwara and T. Kasami, The weight distributions of (256, k) extended binary primitive BCH codes with k ≤ 63 and k ≥ 207. Technical Report of IEICE, IT97–46 (1997–09) (1997) pp. 29–33.
S. M. Johnson, Improved asymptotic bounds for error correcting codes, IEEE Trans. on Inf. Th., Vol. 9, July (1963) pp. 198–205.
T. Kasami, T. Fujiwara and S. Lin, An approximation to the weight distribution of binary linear codes, IEEE Trans. on Inf. Th., Vol. 31, No. 6, November (1985) pp. 769–780.
T. Kasami and S. Lin, Some results on the minimum weight of BCH codes, IEEE Trans. on Inf. Th., Vol. 18, November (1972) pp. 824–825.
I. Krasikov and S. Litsyn, On the distance distribution of duals of BCH codes, IEEE Trans. on Inf. Th., Vol. 45, No. 1, January (1999) pp. 247–250.
T. Laihonen and S. Litsyn, On upper bounds for minimum distance and covering radius for nonbinary codes, Designs, Codes and Cryptography, Vol. 14, No. 1 (1998) pp. 71–80.
J. S. Leon, A probabilistic algorithm for computing minimum weights of large error-correcting codes, IEEE Trans. on Inf. Th., Vol. 34, No. 5, September (1988) pp. 1354–1359.
F. Levy-dit-Vehel, Bounds on the minimum distance of the duals of extended BCH codes, Appl. Alg. Eng. Comm. Comput., Vol. 6, No. 3 (1995) pp. 175–190.
S. Lin, An Introduction to Error-Correcting Codes, Englewood Cliffs, NJ, Prentice Hall (1970).
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland (1977).
MAGMA web page: http://www.maths.usyd.edu.au:8000/u/magma/
MEDICIS web page: http://www.medicis.polytechnique.fr
O. Moreno and C. J. Moreno, The MacWilliams–Sloane conjecture on the tightness of the Carlitz– Uchiyama bound and the weights of duals of BCH codes, IEEE Trans. on Inf. Th., Vol. 40, No. 6, November (1994) pp. 1894–1907.
W. W. Peterson and E. J. Weldon, Jr., Error Correcting Codes, MIT Press (1972).
M. Plotkin, Binary codes with specified minimum distance, IEEE Trans. on Inf. Th., Vol. 6, September (1960) pp. 445–450.
F. Rodier, On the spectra of the duals of binary BCH codes of designed distance delta = 9, IEEE Trans. on Inf. Th., Vol. 38, No. 2, March (1992) pp. 478–479.
M. Sala, Cyclic codes and Gröbner bases, to appear.
M. Sala and A. Tamponi, A linear programming estimate of the weight distribution of BCH(255, k), IEEE Trans. on Inf. Th., Vol. 46, No. 6, September (2000) pp. 2235–2237.
M. Sala and F. Ponchio, A lower bound on the distance of cyclic codes, to appear.
J. Stern, A method for finding codewords of small weight, Coding Theory and Applications, n. 388 in Lecture Notes in Computer Science, Springer Verlag (1989) pp. 106–113.
N. Wax, An upper bound for error detecting and error correcting codes of finite length, IEEE Trans. on Inf. Th., Vol. 5, December (1959) pp. 168–174.
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Sala, M. Upper Bounds on the Dual Distance of BCH(255, k). Designs, Codes and Cryptography 30, 159–168 (2003). https://doi.org/10.1023/A:1025428720732
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DOI: https://doi.org/10.1023/A:1025428720732