A005864 -id:A005864 - cp's OEIS frontend

A005866 The coding-theoretic function A(n,8).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 16, 32, 36, 64, 128, 256, 512, 1024, 2048, 4096

Offset: 1

N. J. A. Sloane

Since A(n,7) = A(n+1,8), A(n,7) gives essentially the same sequence.

The next term, A(25,8), is known to be at least 4096 and at most 5421. - Moshe Milshtein, Dec 03 2018

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).

A. E. Brouwer, Small binary codes: Table of general binary codes, personal web page.
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.
Patric R. J. Östergård, On the Size of Optimal Three-Error-Correcting Binary Codes of Length 16, IEEE Transactions on Information Theory, Volume 57, Issue 10, Oct. 2011.
N. J. A. Sloane and D. S. Whitehead, A New Family of Single-Error Correcting Codes [Shows a(18) >= 36.]
Eric Weisstein's World of Mathematics, Error-Correcting Code.
Index entries for sequences related to A(n,d)

Cf. A005864: A(n,4) and A(n,3), A005865: A(n,6) and A(n,5).

Cf. A005851: A(n,8,5), A005852: A(n,8,6), A005853: A(n,8,7), A004043: A(n,8,8).

a(18)-a(24) from Moshe Milshtein, Dec 03 2018

A005865 The coding-theoretic function A(n,6).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, 24, 32, 64, 128, 256

Offset: 1

N. J. A. Sloane

Since A(n,5) = A(n+1,6), A(n,5) gives essentially the same sequence.

The next term is known only to be in the range 258-340. - Moshe Milshtein, Apr 24 2019

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).

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.
M. Grassl, Bounds on the minimum distance of linear codes
Moshe Milshtein, A new two-error-correcting binary code of length 16, Cryptography and Communications (2019).
Eric Weisstein's World of Mathematics, Error-Correcting Code.
Index entries for sequences related to A(n,d)

Cf. A005864, A005866, A169761, A169762.

A230380 The size of an optimal binary code of length n and edit distance 4.

Original entry on oeis.org

1, 2, 2, 4, 5, 9, 13, 21

Offset: 3

Sheridan Houghten, Oct 17 2013

The edit distance between two words u and v is defined as the minimum number of deletions, insertions, or substitutions required to change u to v.

a(11) >= 32, a(12) >= 54. These lower bounds improve those for a(n) = E_2(n,4) given by the Houghten link (as of March 2024). Examples of codes attaining these bounds are (representing the binary strings as decimal numbers) {0, 63, 85, 150, 243, 252, 263, 281, 356, 417, 479, 538, 610, 680, 887, 915, 924, 1052, 1127, 1136, 1405, 1429, 1446, 1539, 1631, 1644, 1701, 1808, 1935, 1988, 2033, 2046} and {0, 51, 86, 207, 249, 269, 353, 378, 404, 612, 664, 831, 851, 897, 989, 1091, 1128, 1271, 1365, 1502, 1551, 1554, 1628, 1910, 1943, 1956, 2019, 2040, 2076, 2117, 2160, 2321, 2399, 2502, 2583, 2650, 2690, 2937, 3050, 3161, 3198, 3204, 3431, 3483, 3489, 3564, 3585, 3691, 3741, 3752, 3870, 3888, 4032, 4095}, respectively. These codes were found by a simulated annealing search, with the objective function given by the cardinality of a set of binary strings minus a penalty factor times the number of pairs in the set at distance less than 4. - Pontus von Brömssen, Mar 20 2024

Sheridan Houghten, A Table of Bounds on Optimal Fixed-Length Binary Edit-Metric Codes

Cf. A005864, A179183, A230381.

A355226 Irregular triangle read by rows where T(n,k) is the number of independent sets of size k in the n-halved cube graph.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 8, 4, 1, 16, 40, 1, 32, 256, 480, 120, 1, 64, 1344, 11200, 36400, 40320, 13440, 1920, 240, 1, 128, 6336, 156800, 2104480, 15644160, 63672000, 136970880, 147748560, 76396800, 21087360, 4273920, 840000, 161280, 28800, 3840, 240

Offset: 1

Christopher Flippen, Jun 24 2022

The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-halved graph has alpha(G) = A005864(n). The independence polynomial for the n-halved cube is given by Sum_{k=0..alpha(G)} T(n,k)*t^k.

Since 0 <= k <= alpha(G), row n has length A005864(n) + 1.

			Triangle begins:
    k = 0   1   2
n = 1:  1,  1
n = 2:  1,  2
n = 3:  1,  4
n = 4:  1,  8,  4
n = 5:  1, 16, 40
The 4-halved cube graph has independence polynomial 1 + 8*t + 4*t^2.

Eric Weisstein's World of Mathematics, Independence polynomial
Eric Weisstein's World of Mathematics, Halved cube graph

Row sums are A288943.

Cf. A005864, A355558.

from sage.graphs.independent_sets import IndependentSets
from collections import Counter
def row(n):
    if n == 1:
        g = graphs.CompleteGraph(1)
    else:
        g = graphs.HalfCube(n)
    setCounts = Counter()
    for Iset in IndependentSets(g):
        setCounts[len(Iset)] += 1
    outList = [0] * len(setCounts)
    for n in range(0,len(setCounts)):
        outList[n] = setCounts[n]
    return outList

A355558 The independence polynomial of the n-halved cube graph evaluated at -1.

Original entry on oeis.org

0, -1, -3, -3, 25, -135, -2079, 1879969

Offset: 1

Christopher Flippen, Jul 06 2022

The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-halved graph has alpha(G) = A005864(n). The independence polynomial for the n-halved cube is given by Sum_{k=0..alpha(G)} A355226(n,k)*t^k, meaning that a(n) is the alternating sum of row n of A355226.

			Row 5 of A355226 is 1, 16, 40. This means the 5-halved cube graph has independence polynomial 1 + 16*t + 40*t^2. Taking the alternating row sum of row 5, or evaluating the polynomial at -1, gives us 1 - 16 + 40 = 25 = a(5).

Eric Weisstein's World of Mathematics, Independence polynomial
Eric Weisstein's World of Mathematics, Halved cube graph

Cf. A005864, A355226, A288943.

from sage.graphs.independent_sets import IndependentSets
def a(n):
    if n == 1:
        g = graphs.CompleteGraph(1)
    else:
        g = graphs.HalfCube(n)
    icount=0
    for Iset in IndependentSets(g):
        if len(Iset) % 2 == 0:
            icount += 1
        else:
            icount += -1
    return icount

cp's OEIS Frontend

A005866 The coding-theoretic function A(n,8).

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A005865 The coding-theoretic function A(n,6).

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A230380 The size of an optimal binary code of length n and edit distance 4.

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A355226 Irregular triangle read by rows where T(n,k) is the number of independent sets of size k in the n-halved cube graph.

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Sage

A355558 The independence polynomial of the n-halved cube graph evaluated at -1.

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Sage