Abstract
We use asymptotic estimates on coefficients of generating functions to derive anew the asymptotic behaviour of the volume of Hamming spheres and Lee spheres for small alphabets. We then derive the asymptotic volume of Lee spheres for large alphabets, and an asymptotic relation between the covering radius and the dual distance of binary codes.
This work was partially performed while this author was visiting Brown University, Providence, R.I., U.S.A. The author also acknowledges the support of the PRC Mathématique-Informatique (CNRS) and of ESPRIT-II Basic Research Action No. 3075 (project ALCOM).
This work was partially supported by the PRC C3 (CNRS and MRT).
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References
J. Astola, “On the Asymptotic Behaviour of Lee codes,” Discr. Appl. Math, Vol. 8, pp. 13–23 (1984).
E.R. Berlekamp, Algebraic Coding Theory, Aegean Park Press (1984).
H. Cartan, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann (1961).
H.E. Daniels, “Saddlepoint Approximation in Statistics,” Ann. Math. Stat., Vol. 25, pp. 631–650 (1954).
P. Henrici, Applied and Computational Analysis, Wiley (1977).
D. Gardy, Bases de données, allocations aléatoires: quelques analyses de performances, Thèse d'Etat, Université Paxis-Sud, Orsay (1989).
I.J. Good, “Saddle point methods for the multinomial distribution,” Ann. Math. Stat., Vol. 28, pp. 861–881 (1957).
D.H. Greene, D.E. Knuth, Mathematics for the analysis of algorithms, Birkhäuser Verlag (1982).
W.K. Hayman, “A generalisation of Stirling's formula,” Journal für die reine und angewandte Mathematik, Vol. 196, pp. 67–95 (1956).
P. Solé, K. G. Mehrothra, “A Generalization of the Norse Bound to Codes of Higher Strength,” IEEE Trans. Information Theory, IT-37, pp. 190–192 (1991).
A. Tietäväinen, “An Upper Bound on the Covering Radius as a Function of the Dual Distance,” IEEE Trans. Information Theory, IT-36, pp. 1472–1474 (1990).
J. H. van Lint, Introduction to Coding Theory, Springer, Graduate Texts in Math. 86 (1982).
H.S. Wilf, Generatingfunctionology, Academic Press (1990).
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© 1992 Springer-Verlag Berlin Heidelberg
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Gardy, D., Solé, P. (1992). Saddle point techniques in asymptotic coding theory. In: Cohen, G., Lobstein, A., Zémor, G., Litsyn, S. (eds) Algebraic Coding. Algebraic Coding 1991. Lecture Notes in Computer Science, vol 573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0034343
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DOI: https://doi.org/10.1007/BFb0034343
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