A000983
Size of minimal binary covering code of length n and covering radius 1.
Original entry on oeis.org
1, 2, 2, 4, 7, 12, 16, 32, 62
Offset: 1
- 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).
- 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
A029866
Size of minimal binary covering code of length n and covering radius 2.
Original entry on oeis.org
1, 2, 2, 2, 4, 7, 12, 16
Offset: 2
- G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 166.
- 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(9)=16, bounds for n>9]
- Dmitry Kamenetsky, Best known solutions for n <= 11.
- Eric Weisstein's World of Mathematics, Domination Number
- Eric Weisstein's World of Mathematics, Halved Cube Graph
- Index entries for sequences related to covering codes
Cf.
A000983 (domination number of the n-hypercube graph Q_n).
A060439
Triangle T(n,k), 1 <= k <= n, giving maximal size of ternary code of length n and covering radius k.
Original entry on oeis.org
1, 3, 1, 5, 3, 1, 9, 3, 3, 1, 27, 8, 3, 3, 1
Offset: 1
Triangle starts:
1;
3,1;
5,3,1;
9,3,3,1;
...
- G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 174.
A060440
Triangle T(n,k), 1 <= k <= n, giving maximal size of code of length n and covering radius k over alphabet of size 4.
Original entry on oeis.org
1, 4, 1, 8, 4, 1, 24, 7, 4, 1, 64, 16, 4, 4, 1
Offset: 1
1; 4,1; 8,4,1; 24,7,4,1; ...
- G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 174.
A230014
Triangle E(n,k), 1<=k<=n, giving the cardinality of optimal binary covering codes of empty spheres of length n and radius k.
Original entry on oeis.org
2, 2, 4, 4, 4, 8, 4, 4, 4, 16, 8, 6, 6, 8, 32, 14, 8, 6, 8, 14, 64, 24, 8, 8, 8, 8, 24, 128, 32, 16, 8, 8, 8, 16, 32, 256, 64, 24, 12, 10, 10, 12, 24, 64, 512, 124
Offset: 1
Triangle starts:
01: 2,
02: 2, 4,
03: 4, 4, 8,
04: 4, 4, 4, 16,
05: 8, 6, 6, 8, 32,
06: 14, 8, 6, 8, 14, 64,
07: 24, 8, 8, 8, 8, 24, 128,
08: 32, 16, 8, 8, 8, 16, 32, 256,
09: 64, 24, 12, 10, 10, 12, 24, 64, 512,
10: 124, ...
Related to
A060438, which has a code consisting of filled spheres instead of empty spheres.
Related to
A238305, the triangle giving the cardinality of optimal ternary covering codes of empty spheres.
The first column is equal to 2*
A000983.
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