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

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Problemy Peredachi Informatsii, 1978, Volume 14, Issue 1, Pages 3–25 (Mi ppi1518)

This article is cited in 27 scientific papers (total in 27 papers)

Information Theory

On Bounds for Packings on a Sphere and in Space

G. A. Kabatiansky, V. I. Levenshtein

Full-text PDF (1461 kB) Citations (27)

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, Θ)

$M(n,\Theta)$ of points of a unit sphere of

n

$n$ -dimensional Euclidean space at an angular distance of not less than

Θ

$\Theta$ from one another, the method is used to obtain an upper bound that is better than the available ones for any fixed

Θ (0 < Θ < π / 2)

$\Theta\,(0<\Theta<\pi/2)$ and

n \to \infty

$n\to\infty$ This bound yields a new asymptotic upper bound for dn, namely, the maximum packing density of an

n

$n$ -dimensional Euclidean space by equal balls.

Received: 26.01.1977

Bibliographic databases:

Document Type: Article

UDC: 621.391.1:519

Language: Russian

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://mathscinet.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

Linking options:

https://www.mathnet.ru/eng/ppi1518

https://www.mathnet.ru/eng/ppi/v14/i1/p3

This publication is cited in the following 27 articles:

Afkhami-Jeddi N. Cohn H. Hartman T. de Laat D. Tajdini A., “High-Dimensional Sphere Packing and the Modular Bootstrap”, J. High Energy Phys., 2020, no. 12, 66
Izv. Math., 83:3 (2019), 540–564
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
G. K. Kamenev, “Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls”, Comput. Math. Math. Phys., 56:5 (2016), 744–755
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
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
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
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
Proc. Steklov Inst. Math., 275 (2011), 229–238
Proc. Steklov Inst. Math., 263 (2008), 134–149
Ben-Haim, Y, “Improved upper bounds on the reliability function of the Gaussian channel”, IEEE Transactions on Information Theory, 54:1 (2008), 5
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
M. V. Burnashev, “Code Spectrum and the Reliability Function: Gaussian Channel”, Problems Inform. Transmission, 43:2 (2007), 69–88
A. M. Raigorodskii, “On a problem in the geometry of numbers”, Tr. In-ta matem., 15:1 (2007), 111–117
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
A. M. Barg, D. Yu. Nogin, “Spectral Approach to Linear Programming Bounds on Codes”, Problems Inform. Transmission, 42:2 (2006), 77–89
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
M. A. Vsemirnov, M. G. Rzhevskii, “An upper bound for the contact number in dimension 9”, Russian Math. Surveys, 57:5 (2002), 1015–1016
N. N. Andreev, “A minimal design of order $11$ on the $3$ -sphere”, Math. Notes, 67:4 (2000), 417–424
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

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

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