Upper bounds on the minimum distance of spherical codes

IEEE Transactions on Information Theory

Volumn 42, Issue 5, 1996, Pages 1576-1581

  Boyvalenkov, Peter G ^a   Danev, Danyo P ^a   Bumova, Silvya P ^a  

^a INSTITUTE OF MATHEMATICS AND INFORMATICS   (Bulgaria)

Author keywords

Gegeubauer polynomials; Minimum distance; Spherical codes

Indexed keywords

ALGORITHMS; CALCULATIONS; COMBINATORIAL MATHEMATICS; LINEAR PROGRAMMING; NUMBER THEORY; POLYNOMIALS;

GEGENBAUER POLYNOMIALS; LEVENSHTEIN BOUNDS; SPHERICAL CODES;

CODES (SYMBOLS);

EID: 0030242504     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.532903     Document Type: Article

References (31)

1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1965.
2. R. Askey, "Orthogonal polynomials and special functions," in SIAM Reg. Conf. Lecttures on Applied Mathematics, vol. 21, 1975.
3. J. T. Astola, "The Tietäväinen bound for spherical codes," Discr. Appl. Math., vol. 7, pp. 17-21, 1984.
4. E. Bannai and N. J. A. Sloane, "Uniqueness of certain spherical codes," Canad. J. Math., vol. 33, pp. 437-449, 1981.
5. P. G. Boyvalenkov, "Bounds on packings in Euclidean spaces," Ph.D. dissertation, Center for Informatics and Computer Technology, Sofia, Bulgaria, 1993.
6. _, "Extremal polynomials for obtaining bounds for spherical codes and designs," Discr. Comp. Geom. vol. 14, pp. 167-183, 1995.
7. P. G. Boyvalenkov and I. N. Landgev, "On maximal spherical codes I," in Lecture Notes in Computer Science, vol. 948, pp. 158-168, 1995.
8. P. G. Boyvalenkov, D. Danev, and I. N. Landgev, "On maximal spherical codes II," submitted for publication.
9. P. G. Boyvalenkov, D. Danev, and S. Dimcheva, "On bounds on the minimum distance of spherical codes," in Intern. Workshop on Optimal Codes (Sozopol, Bulgaria, 1995), pp. 25-28.
10. K. Böröczky, "Packing of spheres in spaces of constant curvature," Acta Math. Acad. Scien. Hungar., vol. 32, pp. 243-261, 1978.
11. A. R. Calderbank, J. E. Mazo, and V. K. Wei, "Asymptotic upper bounds on the minimum distance of trellis codes," IEEE Trans. Inform. Theory, vol. IT-28, pp. 55-67, Jan. 1982.
12. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. New York: Springer-Verlag, 1988.
13. B. W. Clare and D. L. Kepert, "The closest packing of equal circles on a sphere," Proc. Roy. Soc. Lond., vol. 405, pp. 329-344, 1986.
14. L. Danzer, "Endliche Punktmengen auf der 2-Sphäre mit möglichst großem Minimalabstand," Habilitationsschrift (Ph.D. dissertation), Univ. Göttingen, Göttingen, Germany, 1965.
15. P. Delsarte, J.-M.Goethals, and J. J. Seidel, "Spherical codes and designs," Geom. Dedicata, vol. 6, pp. 363-388, 1977.
16. T. Ericson and V. Zinoviev, "Best spherical codes," 1994, preprint.
17. _, "Spherical codes generated by binary partitions of symmetric pointsets," IEEE Trans. Inform. Theory, vol. 41, pp. 107-129, Jan. 1995.
18. _, "Spherical codes," in preparation.
19. L. Fejes Töth, "Über die Abscätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystemes," Jber. Deut. Math. Verein., vol. 53, pp. 66-68, 1943.
20. G. A. Kabatianskii and V. I. Levenshtein, "Bounds for packings on a sphere and in space," Probl. Inform. Transm., vol. 14, pp. 1-17, 1978.
21. V. I. Levenshtein, "On bounds for packings in n-dimensional Euclidean space," Sov. Math.-Dokl., vol. 20, pp. 417-421, 1979.
22. _, "Bounds for packings in metric spaces and certain applications," Probl. Kibernetiki, vol. 40, pp. 44-110, 1983 (in Russian).
23. _, "Designs as maximum codes in polynomial metric spaces, " Acta Applicandae Math., vol. 25, pp. 1-83, 1992.
24. A. M. Odlyzko and N. J. A. Sloane, "New bounds on the number of unit spheres that can touch a unit sphere in n dimensions," J. Comb. Theory A, vol. 26, pp. 210-214, 1979.
25. G. J. Pottie and D. P. Taylor, "Sphere-packing upper bounds on the free distance of trellis codes," IEEE Trans. Inform. Theory, vol. 34, pp. 435-447, 1988.
26. R. M. Robinson, "Arrangements of 24 points on a sphere," Math. Ann., vol. 144, pp. 17-48, 1961.
27. K. Schütte and B. L. van der Waerden, "Auf welcher Kugel haben 5, 6, 7 oder 8 Punkte mit Mindestabsland Eins Platz?," Math. Ann., vol. 123, pp. 96-124, 1951.
28. V. M. Sidel'nikov, "On the density of sphere packings on the surface of an n-dimensional Euclidean sphere and on the cardinality of a binary code with a given minimum distance," Sov. Math.-Dokl., vol. 14, pp. 1851-1855, 1973.
29. _, "New bounds for densest packing of spheres in n-dimensional Euclidean space," Matem. Sbornik, vol. 95, pp. 148-158, 1974 (in Russian).
30. _, "On extremal polynomials used to estimate the size of codes," Probl. Inform. Transm., vol. 16, pp. 174-186, 1980.
31. A. Tietäväinen, "Bounds for binary codes just outside the Plotkin range," Inform. Contr., vol. 47, pp. 85-93, 1980.