A005864 The coding-theoretic function A(n,4). (Formerly M1111)

1, 1, 1, 2, 2, 4, 8, 16, 20, 40, 72, 144, 256, 512, 1024, 2048

Offset

1,4

Comments

Since A(n,3) = A(n+1,4), A(n,3) gives essentially the same sequence.

The next term a(17) is in the range 2816-3276.

Let T_n be the set of SDS-maps of sequential dynamical systems defined over the complete graph K_n in which all vertices have the same vertex function (defined using a set of two possible vertex states). Then a(n) is the maximum number of period-2 orbits that a function in T_n can have. - Colin Defant, Sep 15 2015

Since the n-halved cube graph is isomorphic to (or, if you prefer, defined as) the graph with binary sequences of length n-1 as nodes and edges between pairs of sequences that differ in at most two positions, the independence number of the n-halved cube graph is A(n-1,3) = a(n). - Pontus von Brömssen, Dec 12 2018

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 248.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 674.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..16.

A. E. Brouwer, Tables of general binary codes

A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.

Colin Defant, Binary Codes and Period-2 Orbits of Sequential Dynamical Systems, arXiv:1509.03907 [math.CO], 2015.

Moshe Milshtein, A new binary code of length 16 and minimum distance 3, Information Processing Letters 115.12 (2015): 975-976.

Patric R. J. Östergård (patric.ostergard(AT)hut.fi), T. Baicheva and E. Kolev, Optimal binary one-error-correcting codes of length 10 have 72 codewords, IEEE Trans. Inform. Theory, 45 (1999), 1229-1231.

A. M. Romanov, New binary codes with minimal distance 3, Problemy Peredachi Informatsii, 19 (1983) 101-102.

Eric Weisstein's World of Mathematics, Error-Correcting Code

Eric Weisstein's World of Mathematics, Halved Cube Graph

Eric Weisstein's World of Mathematics, Independence Number

Wikipedia, Halved cube graph

Index entries for sequences related to A(n,d)

Crossrefs

Cf. A005865: A(n,6) ~ A(n,5), A005866: A(n,8) ~ A(n,7).

Cf. A001839: A(n,4,3), A001843: A(n,4,4), A169763: A(n,4,5).

Sequence in context: A279059 A054243 A289670 * A112433 A171648 A189914

Adjacent sequences: A005861 A005862 A005863 * A005865 A005866 A005867

Keyword

nonn,hard,nice,more

Author

N. J. A. Sloane

Status

approved