Problems of Information Transmission

, Volume 54, Issue 1, pp 48–55 | Cite as

On Metric Dimension of Nonbinary Hamming Spaces

  • G. A. KabatianskyEmail author
  • V. S. Lebedev
Coding Theory


For q-ary Hamming spaces we address the problem of the minimum number of points such that any point of the space is uniquely determined by its (Hamming) distances to them. It is conjectured that for a fixed q and growing dimension n of the Hamming space this number asymptotically behaves as 2n/ log q n. We prove this conjecture for q = 3 and q = 4; for q = 2 its validity has been known for half a century.


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  1. 1.
    Harary, F. and Meter, R.A., On the Metric Dimension of a Graph, Ars Combin., 1976, vol. 2, pp. 191–195.Google Scholar
  2. 2.
    Slater, P., Leaves on Trees, Proc. 6th Southeastern Conf. on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, Feb. 17–20, 1975), Hoffman, F., Mullin, R.C., Levow, R.B., Roselle, D., Stanton, R.G., and Thomas, R.S.D., Eds., Congr. Numer., vol. XIV, Winnipeg: Utilitas Math., 1975, pp. 549–599.Google Scholar
  3. 3.
    Erdős, P. and Rényi, A., On Two Problems of Information Theory, Magyar Tud. Akad. Mat. Kutató Int. Közl., 1963, vol. 8, pp. 241–254.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Erdős, P. and Spencer, J., Probabilistic Methods in Combinatorics, Budapest: Acad. Kiadó, 1974. Translated under the title Veroyatnostnye metody v kombinatorike, Moscow: Mir, 1976.zbMATHGoogle Scholar
  5. 5.
    Lindström, B., On a Combinatory Detection Problem. I, Magyar Tud. Akad. Mat. Kutató Int. Közl., 1964, vol. 9, pp. 195–207.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cantor, D.G. and Mills, W.H., Determination of a Subset from Certain Combinatorial Properties, Canad. J. Math., 1966, vol. 18, pp. 42–48.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chvatal, V., Mastermind, Combinatorica, 1983, vol. 3, no. 3–4, pp. 325–329.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kabatianski, G., Lebedev, V., and Thorpe, J., The Mastermind Game and the Rigidity of the Hamming Space, in Proc. 2000 IEEE Int. Sympos. on Information Theory (ISIT’2000), Sorrento, Italy, June 25–30, 2000, p.375.Google Scholar
  9. 9.
    Du, D.-Z. and Hwang, F.K., Combinatorial Group Testing and Its Applications, Singapore: World Sci., 2000, 2nd ed.zbMATHGoogle Scholar
  10. 10.
    Chang, S.C. and Weldon, E.J., Jr., Coding for T -User Multiple-Access Channels, IEEE Trans. Inform. Theory, 1979, vol. 25, no. 6, pp. 684–691.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martirosian, S.S. and Khachatrian, G.G., Construction of Signature Codes and the Coin Weighing Problem, Probl. Peredachi Inf., 1989, vol. 25, no. 4, pp. 96–97 [Probl. Inf. Trans. (Engl. Transl.), 1989, vol. 25, no. 4, pp. 334–335].Google Scholar
  12. 12.
    Lindström, B., On a Combinatorial Problem in Number Theory, Canad. Math. Bull., 1965, vol. 8, no. 4, pp. 477–490.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hoeffding, W., Probability Inequalities for Sums of Bounded Random Variables, J. Amer. Statist. Assoc., 1963, vol. 58, no. 301, pp. 13–30.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.Skolkovo Institute of Science and TechnologyMoscowRussia
  2. 2.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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