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Probl. Peredachi Inf., 1978, Volume 14, Issue 1, Pages 3–25 (Mi ppi1518)  

This article is cited in 26 scientific papers (total in 26 papers)

Information Theory

On Bounds for Packings on a Sphere and in Space

G. A. Kabatiansky, V. I. Levenshtein


Abstract: A method is proposed for obtaining bounds for packings in metric spaces, the method being based on the use of zonal spherical functions associated with a motion group of the space. For the maximum number $M(n,\Theta)$ of points of a unit sphere of $n$-dimensional Euclidean space at an angular distance of not less than $\Theta$ from one another, the method is used to obtain an upper bound that is better than the available ones for any fixed $\Theta (0<\Theta<\pi/2)$ and $n\to\infty$ This bound yields a new asymptotic upper bound for dn, namely, the maximum packing density of an $n$-dimensional Euclidean space by equal balls.

Full text: PDF file (1461 kB)

English version:
Problems of Information Transmission, 1978, 14:1, 1–17

Bibliographic databases:

UDC: 621.391.1:519
Received: 26.01.1977

Citation: G. A. Kabatiansky, V. I. Levenshtein, “On Bounds for Packings on a Sphere and in Space”, Probl. Peredachi Inf., 14:1 (1978), 3–25; Problems Inform. Transmission, 14:1 (1978), 1–17

Citation in format AMSBIB
\Bibitem{KabLev78}
\by G.~A.~Kabatiansky, V.~I.~Levenshtein
\paper On Bounds for Packings on a~Sphere and in Space
\jour Probl. Peredachi Inf.
\yr 1978
\vol 14
\issue 1
\pages 3--25
\mathnet{http://mi.mathnet.ru/ppi1518}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=514023}
\zmath{https://zbmath.org/?q=an:0407.52005}
\transl
\jour Problems Inform. Transmission
\yr 1978
\vol 14
\issue 1
\pages 1--17


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. S. Ryshkov, E. P. Baranovskii, “Classical methods in the theory of lattice packings”, Russian Math. Surveys, 34:4 (1979), 1–68  mathnet  crossref  mathscinet  zmath
    2. S. N. Litsyn, M. A. Tsfasman, “Algebro-geometrical and number-theoretical packings of balls in $\mathbb R^N$”, Russian Math. Surveys, 40:2 (1985), 219–220  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. G. Babenko, “An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces”, Izv. Math., 62:6 (1998), 1095–1119  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. P. Boyvalenkov, D. Danev, “On Linear Programming Bounds for Codes in Polynomial Metric Spaces”, Problems Inform. Transmission, 34:2 (1998), 108–120  mathnet  mathscinet  zmath
    5. N. N. Andreev, “A spherical code”, Russian Math. Surveys, 54:1 (1999), 251–253  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. Reuven, I, “The weighted coordinates bound and trellis complexity of block codes and periodic packings”, IEEE Transactions on Information Theory, 45:5 (1999), 1658  crossref  isi
    7. N. N. Andreev, “A minimal design of order $11$ on the $3$-sphere”, Math. Notes, 67:4 (2000), 417–424  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. V. V. Arestov, A. G. Babenko, “Estimates of the maximal value of angular code distance for 24 and 25 points on the unit sphere in $\mathbb R^4$”, Math. Notes, 68:4 (2000), 419–435  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. M. V. Burnashev, “On the Relation between the Code Spectrum and the Decoding Error Probability”, Problems Inform. Transmission, 36:4 (2000), 285–304  mathnet  mathscinet  zmath
    10. M. A. Vsemirnov, M. G. Rzhevskii, “An upper bound for the contact number in dimension 9”, Russian Math. Surveys, 57:5 (2002), 1015–1016  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    11. A. M. Barg, D. Yu. Nogin, “Spectral Approach to Linear Programming Bounds on Codes”, Problems Inform. Transmission, 42:2 (2006), 77–89  mathnet  crossref  mathscinet
    12. Ben-Haim Ya., Litsyn S., “Improved upper bounds on the reliability function of the Gaussian channel”, 2006 IEEE International Symposium on Information Theory, 2006, 709–713  crossref  isi
    13. M. V. Burnashev, “Code Spectrum and the Reliability Function: Gaussian Channel”, Problems Inform. Transmission, 43:2 (2007), 69–88  mathnet  crossref  mathscinet  zmath  isi
    14. A. M. Raigorodskii, “On a problem in the geometry of numbers”, Tr. In-ta matem., 15:1 (2007), 111–117  mathnet
    15. Burnashev M.V., “New Results on the Reliability Function of the Gaussian Channel”, 2007 IEEE International Symposium on Information Theory Proceedings, Vols 1-7, IEEE, 2007, 471–474  crossref  isi
    16. Proc. Steklov Inst. Math., 263 (2008), 134–149  mathnet  crossref  mathscinet  zmath  isi  elib
    17. Ben-Haim, Y, “Improved upper bounds on the reliability function of the Gaussian channel”, IEEE Transactions on Information Theory, 54:1 (2008), 5  crossref  isi
    18. Barg A., Nogin D., “A Functional View of Upper Bounds on Codes”, Coding and Cryptology, Series on Coding Theory and Cryptology, 4, eds. Li Y., Ling S., Niederreiter H., Wang H., Xing C., Zhang S., World Scientific Publ Co Pte Ltd, 2008, 15–24  crossref  isi
    19. N. A. Kuklin, “Vid ekstremalnoi funktsii v zadache Delsarta otsenki sverkhu kontaktnogo chisla trekhmernogo prostranstva”, Tr. IMM UrO RAN, 17, no. 3, 2011, 225–232  mathnet  elib
    20. Proc. Steklov Inst. Math., 275 (2011), 229–238  mathnet  crossref  mathscinet  isi  elib  elib
    21. N. A. Kuklin, “Delsarte method in the problem on kissing numbers in high-dimensional spaces”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 108–123  mathnet  crossref  isi  elib
    22. N. A. Kuklin, “The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 99–111  mathnet  crossref  mathscinet  isi  elib
    23. G. K. Kamenev, “Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality”, Comput. Math. Math. Phys., 54:8 (2014), 1201–1213  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    24. G. K. Kamenev, “Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls”, Comput. Math. Math. Phys., 56:5 (2016), 744–755  mathnet  crossref  crossref  isi  elib
    25. Izv. Math., 83:3 (2019), 540–564  mathnet  crossref  crossref  adsnasa  isi  elib
    26. Boyvalenkov P.G., Dragnev P.D., Hardin D.P., Saff E.B., Stoyanova M.M., “Energy Bounds For Codes in Polynomial Metric Spaces”, Anal. Math. Phys., 9:2, SI (2019), 781–808  crossref  mathscinet  zmath  isi  scopus
  • Проблемы передачи информации Problems of Information Transmission
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