Abstract
We estimate the interval where the distance distribution of a code of length n and of given dual distance is upperbounded by the binomial distribution. The binomial upper bound is shown to be sharp in this range in the sense that for every subinterval of size about √n ln n there exists a spectrum component asymptotically achieving the binomial bound. For self-dual codes we give a better estimate for the interval of binomiality.
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References
Th. Beth, H. Kalouti and D. E. Lazic, Which families of long binary linear codes have a binomial weight distribution? Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, G. Cohen, M. Giusti and T. Mora, Eds., Lecture Notes in Computer Science, v. 948, Springer, pp. 120-130.
Th. Beth, D. E. Lazic and V. Senk, A family of binary codes with asymptotically good distance distribution, International Symposium on Coding Theory and Applications, Udine, Italy,Proceedings, Springer-Verlag, (1990).
E. L. Blokh and V.V. Zyablov, Linear Concatenated CodesMoscow, Nauka, (In Russian) (1982).
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, No. 10 (1973).
I. Gashkov and V. Sidelnikov, Linear ternary quasiperfect codes correcting double errors, Probl. Peredachi Inform., Vol. 22, No.4 (1986) pp. 43-48.
T. Kasami, T. Fujiwara and S. Lin, An approximation to the weight distribution of binary linear codes, IEEE Trans. Inform. Theory, Vol. 31, No.6 (1985) pp. 769-780.
I. Krasikov and S. Litsyn, On spectra of BCH codes, IEEE Trans. Inform. Theory, Vol. 41, No.3 (1995) pp. 786-788.
I. Krasikov and S. Litsyn, On the accuracy of the binomial approximation to the distance distribution of codes, IEEE Trans. Inform. Theory, Vol. 41, No.5 (1995) pp. 1472-1475.
V. Levenshtein, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, Vol. 41, No.5 (1995) pp. 1303-1321.
J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, (1992).
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, New York: North-Holland, (1977).
V. M. Sidelnikov, Weight spectrum of binary Bose-Chaudhuri-Hocquenghem codes, Probl. Peredachi Inform., Vol. 7, No.1 (1971) pp. 14-22.
P. Solé, A limit law on the distance distribution of binary codes, IEEE Trans. Inform.Theory, Vol. 36, No.1 (1990) pp. 229-232.
P. Solé, Weight distribution and dual distance, AAECC, vol. 5, No.2 (1994) pp. 117-122.
P. Solé and Ph.Stokes, Covering radius, codimension and dual distance width, IEEE Trans. Inform. Theory, Vol. 39, No.4 (1993) pp. 1195–1203.
A. Tietäväinen, An upper bound on the covering radius as a function of the dual distance, IEEE Trans. Inform. Theory, Vol. 36, No.5 (1990) pp. 1472-1474.
A. Tietäväinen, Covering radius and dual distance, Designs, Codes and Cryptography, Vol. 1, No.1 (1991) pp. 31-46.
H. N. Ward, A bound for divisible codes, IEEE Trans. Inform.Theory, Vol. 38, No.1 (1992) pp. 191-194.
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Krasikov, I., Litsyn, S. Bounds on Spectra of Codes with Known Dual Distance. Designs, Codes and Cryptography 13, 285–297 (1998). https://doi.org/10.1023/A:1008206125050
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DOI: https://doi.org/10.1023/A:1008206125050