A000983 Size of minimal binary covering code of length n and covering radius 1.

1, 2, 2, 4, 7, 12, 16, 32, 62

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

For k > 0, a(2^k-1) = 2^(2^k-k-1). In this case the minimal covering code is also a Hamming code.

In the game described in the Wikipedia link, with n players, the optimal strategy wins with probability 1-a(n)/2^n. In the optimal strategy, the players first agree on a minimal covering code of length n. After the hats are placed, each player knows two words of length n such that one of them is the hat colors of the n players. If one of these two words is a member of the covering code and the other word is not, that player guesses the word that is not. Otherwise, that player does not guess. This strategy guarantees that the team wins for all words that are not members of the covering code.

Because each codeword covers n+1 of the 2^n words, A053637(n+1) is a lower bound. - Rob Pratt, Jan 05 2015

a(n) is also the domination number of the n-hypercube graph Q_n. - Eric W. Weisstein, Feb 20 2016

The next term a(10) is in the range 107-120. - Andrey Zabolotskiy, Sep 01 2016

References

• G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 166.
• I. S. Honkala and Patric R. J. Östergård, Code design, Chapter 13 of Local Search in Combinatorial Optimization, E. Aarts and J. K. Lenstra (editors), Wiley, New York 1997, pp. 441-456.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Jernej Azarija, M. A. Henning, S. Klavzar, (Total) Domination in Prisms, arXiv preprint arXiv:1606.08143 [math.CO], 2016. See Table 1.
• R. Bertolo, Patric R. J. Östergård and W. D. Weakley, An updated table of binary/ternary mixed covering codes, J. Combin. Designs, 12 (2004), 157-176, DOI:10.1002/jcd.20008. [a(10)>=107]
• H. Hamalainen et al., Football pools - a game for mathematicians, Amer. Math. Monthly, 102 (1995), 579-588.
• J. G. Kalbfleisch and R. G. Stanton, A combinatorial problem in matching, J. London Math. Soc., 44 (1969), 60-64.
• Dmitry Kamenetsky, Best known solutions for n <= 11.
• A. Lobstein, G. Cohen and N. J. A. Sloane, Recouvrements d'Espaces de Hamming Binaires, C. R. Acad. Sci. Paris, Series I, 301 (1985), 135-138.
• Patric R. J. Östergård and Markku K. Kaikkonen, New upper bounds for binary covering codes, Discrete Mathematics 178 (1998), 165-179.
• Patric R. J. Östergård and U. Blass, On the size of optimal binary codes of length 9 and covering radius 1, IEEE Trans. Inform. Theory, 47 (2001), 2556-2557. [Determines a(9)].
• Eric Weisstein's World of Mathematics, Domination Number
• Eric Weisstein's World of Mathematics, Hypercube Graph
• Wikipedia, Hat puzzle (Ebert's version and Hamming codes)
• Index entries for sequences related to covering codes

Crossrefs

A column of A060438. Cf. A029866.