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Non linear covering codes : A few results and conjectures

  • Gérard D. Cohen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 356)

Abstract

Denote by K(n,t) the minimum number of elements in a covering of IF 2 n with Hamming spheres of radius t, and by μn(t) the density of such a covering :
$$\mu _n (t): = 2^{ - n} K(n,t) \mathop \Sigma \limits_{i = 0}^t ( \mathop {}\limits_i^n ).$$
For fixed t, μn (t) is upperbounded by a real number c(t) independent of n. For t=1, one can take c(1)=1.5. We conjecture : Conjecture 1.\(\mathop {lim}\limits_{n \to \infty } \mu _n (1) = 1.\) μn(1)=1. Another conjecture is : Conjecture 2. K(n+2,t+1)≤K(n,t). We prove this for t=1 and discuss a possible way of proving it for higher t, by extensions of the concept of normality.

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References

  1. [1]
    M. BEVERAGGI, G.D. COHEN: "On the density of best coverings in Hamming spaces", in "Coding Theory and Applications", Springer-Verlag Lecture Notes in Computer Science. 311 (G. COHEN, P. GODLEWSKI eds.) 1987, pp. 39–44.Google Scholar
  2. [2]
    G.D. COHEN, A.C. LOBSTEIN, N.J.A. SLOANE: "Further results on the covering radius of codes", IEEE Trans. on Information Theory, vol. IT-32, no 5, Sept. 1986, pp. 680–694.CrossRefGoogle Scholar
  3. [3]
    G.D. COHEN, A.C. LOBSTEIN, N.J.A. SLOANE: "On a conjecture concerning coverings of Hamming space", Lecture Notes in Computer Science 228, 1985, pp. 79–90.Google Scholar
  4. [4]
    I. HONKALA: "Lower bounds for binary covering codes", IEEE Trans. on Information Theory. Vol. IT-34, no2, March 1988, pp. 326–329.CrossRefGoogle Scholar
  5. [5]
    I. HONKALA: "Bounds for binary constant weight and covering codes", Licentiate Thesis, University of Turku, March 1987.Google Scholar
  6. [6]
    R.L. GRAHAM, N.J.A. SLOANE: "On the covering radius of codes", IEEE Trans. on Information Theory, vol. IT-31, no 3, May 1985, pp. 385–401.CrossRefGoogle Scholar
  7. [7]
    K.E. KILBY, N.J.A. SLOANE: "On the covering radius problem for codes (II)" SIAM J. Alg. Discrete Methods. Vol. 8, no4, October 1987, pp. 619–627.Google Scholar
  8. [8]
    Y.G. SALIN: Personal Communication.Google Scholar
  9. [9]
    G.J.M. VAN WEE: "Improved sphere bounds on the covering radius of codes", IEEE Trans. On Information Theory, vol. IT-34, no2, March 1988, pp. 237–245.Google Scholar
  10. [10]
    L.T. WILLE: "The football pool problem for 6 matches: a new upper bound obtained by simulated annealing", J. of Combinatorial theory, Series A, vol. 45, no2, July 1987, pp. 171–177.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Gérard D. Cohen
    • 1
  1. 1.ENST and CNRS UA 820France

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