cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A060438 Triangle T(n,k), 1 <= k <= n, giving maximal size of binary code of length n and covering radius k.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Apr 07 2001

Keywords

Examples

			Triangle starts:
1;
2,1;
2,2,1;
4,2,2,1;
7,2,2,2,1;
...

References

  • G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 166.

Links

Crossrefs

Columns give A000983, A029866. Cf. A060439, A060440.

Extensions

Row 9 from Andrey Zabolotskiy, Apr 11 2017 using Gerzson Kéri's tables.

A238305 Triangle E(n,k), 1<=k<=n, giving the cardinality of optimal ternary covering codes of empty spheres of length n and radius k.

Original entry on oeis.org

2, 3, 3, 6, 4, 5, 14, 6, 5, 8, 27, 12, 6, 7, 12
Offset: 1

Views

Author

Kamiel P.F. Verstraten, Feb 24 2014

Keywords

Comments

The next term is in the range 71-81.
Right diagonal is equal to A086676.

Examples

			Triangle starts:
01: 2
02: 3 3
03: 6 4 5
04: 14 6 5 8
05: 27 12 6 7 12
...

Links

Crossrefs

Related to A060439, which has a code consisting of filled spheres instead of empty spheres.
Related to A230014, the triangle giving the cardinality of optimal binary covering codes of empty spheres.
See also A000983.

A004044 The classic football pool problem: size of minimal covering code in {0,1,2}^n with covering radius 1.

Original entry on oeis.org

1, 1, 3, 5, 9, 27
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The next 3 terms a(6..8) are in the ranges 71-73, 156-186, 402-486. Also a(13) = 3^10 [Kamps and van Lint, 1969].
Because each codeword covers 2n+1 of the 3^n words, ceiling(3^n/(2n+1)) is a lower bound. - Rob Pratt, Jan 06 2015
a((3^m-1)/2) = 3^((3^m-1)/2 - m) follows from the existence of ternary Hamming codes in these dimensions (see page 286 of [Cohen et al.]).
a(n+1) <= 3*a(n): given a covering of {0,1,2}^n, copy it in each of {i}x{0,1,2}^n for i = 0, 1, 2.
Combining the above three comments, one obtains ceiling(3^n/(2n+1)) <= a(n) <= 3^(n-floor(log_3(2n+1))) for n >= 0.
Conjecture: a((3^m+1)/2) = 3^((3^m+1)/2 - m) for m > 0; i.e., a((3^m-1)/2 + 1) = 3 * a((3^m-1)/2) for m > 0. - Thomas Ordowski, Jul 10 2021

Examples

			An example for a(4) = 9 is {0000, 0112, 0221, 1022, 1101,1210, 2011, 2120, 2202}. - _Robert P. P. McKone_, Jun 27 2021
For a(5) = 27, prepend each of these 9 codewords by 0, 1, and 2. - _Rob Pratt_, Jun 27 2021
van Laarhoven et al. (1989) give examples for a(6), a(7), a(8) which are the best presently known. - _R. J. Mathar_, Jun 29 2021

References

  • Cohen, Gérard, Iiro Honkala, Simon Litsyn, and Antoine Lobstein, Covering Codes, North-Holland, 1997, p. 174.
  • H. J. L. Kamps and J. H. van Lint. "A covering problem." In Colloq. Math. Soc. Janos Bolyai; Hungar. Combin. Theory and Appl., Balantonfured, Hungary, pp. 679-685, 1969.

Links

Crossrefs

First column of A060439.

Extensions

Bounds corrected and corresponding reference added by Jan Kristian Haugland, Mar 10 2010
Edited with more references. - N. J. A. Sloane, Jun 21 2021

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

Views

Author

N. J. A. Sloane, Apr 07 2001

Keywords

Examples

			1; 4,1; 8,4,1; 24,7,4,1; ...

References

  • G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 174.

Links

Crossrefs

Cf. A060438, A060439.

Extensions

T(5,2)-T(5,5) from Andrey Zabolotskiy, Apr 11 2017 using Gerzson Kéri's tables.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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