A034254 -id:A034254 - cp's OEIS frontend

A034358 Number of binary [ n,4 ] codes.

0, 0, 0, 1, 5, 16, 43, 106, 240, 516, 1060, 2108, 4064, 7641, 14036, 25253, 44560, 77245, 131658, 220883, 365027, 594674, 955649, 1515908, 2374875, 3676632, 5627587, 8520689, 12767557, 18941641, 27834607, 40530902, 58503994, 83741461, 118904892, 167534794, 234309554, 325373538, 448747606

Offset: 1

N. J. A. Sloane

nonn

Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=4.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [Here a(n) = W_{n,4,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = W_{n,4,2}; see p. 197.]
Petros Hadjicostas, Generating function for a(n).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
M. Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
M. Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Index entries for sequences related to binary linear codes

Cf. A034253, A034254.
Column k=4 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034345.

# Fripertinger's method to find the g.f. of column k >= 2 of A076831 or A034356 (for small k):
def A076831col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = (f1 - f2)/(1-x)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 (this sequence) gives
print(A076831col(4, 30)) # Petros Hadjicostas, Oct 07 2019

More terms from Petros Hadjicostas, Oct 07 2019

A034357 Number of binary [ n,3 ] codes.

Original entry on oeis.org

0, 0, 1, 4, 10, 22, 43, 77, 131, 213, 333, 507, 751, 1088, 1546, 2159, 2967, 4023, 5384, 7122, 9322, 12081, 15512, 19752, 24950, 31283, 38953, 48188, 59244, 72419, 88037, 106469, 128129, 153476, 183019, 217331, 257033

Offset: 1

N. J. A. Sloane

nonn

Also, a(n) is the number of orbits of C_2^3 subgroups of C_2^n under automorphisms of C_2^n. Also, a(n) is the number of faithful representations of C_2^3 of dimension n up to equivalence by automorphisms of (C_2^3). - Andrew Rupinski, Jan 20 2011

H. Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=3.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have a(n) = W_{n,3,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [We have a(n) = W_{n,3,2}; see p. 197.]

Cf. A034198, A034253, A034254.
Column k=3 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034344.

G.f.: (-x^15+2*x^14-x^13+x^12+x^9-x^7+x^4+x^3)/((1-x)^3*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^7)).

A034359 Number of binary [ n,5 ] codes.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 23, 77, 240, 705, 1988, 5468, 14724, 39006, 101818, 261924, 663748, 1655781, 4062110, 9793065, 23186825, 53896597, 122975627, 275449464, 605794093, 1308633243, 2777847319, 5797093774, 11900199553, 24042491094, 47833081481, 93765335118, 181200186060, 345389067067, 649704599010

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=5.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have a(n) = W_{n,5,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [We have a(n) = W_{n,5,2}; see p. 197.]
Petros Hadjicostas, Generating function for a(n).

Cf. A034253, A034254.
Column k=5 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034346.

More terms from Joerg Arndt, Oct 09 2019

A034360 Number of binary [ n,6 ] codes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 32, 131, 516, 1988, 7664, 29765, 117169, 467266, 1880517, 7588675, 30491836, 121191234, 473940269, 1816579108, 6806904522, 24897540538, 88831250408, 309108741706, 1049278764758, 3476233500031, 11246972937210, 35561409388625, 109967835029368, 332834886787933, 986732945823099

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=6.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have a(n) = W_{n,6,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [We have a(n) = W_{n,6,2}; see p. 197.]
Petros Hadjicostas, Generating function for a(n).

Cf. A034253, A034254.
Column k=6 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034347.

More terms from Joerg Arndt, Oct 09 2019

A034361 Number of binary [ n,7 ] codes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 8, 43, 213, 1060, 5468, 29765, 173035, 1074526, 7059804, 48235007, 336048291, 2345912476, 16193974418, 109563962854, 722594600193, 4631590699334, 28811338570224, 173868030213652

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254.
Column k=7 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034348.

A034362 Number of binary [ n,8 ] codes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 9, 56, 333, 2108, 14724, 117169, 1074526, 11249092, 130484439, 1612782351, 20497233072, 260975054461, 3273854883027, 40073904283055, 476142523109291, 5477680380616386, 60959857679340812

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254.
Column k=8 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034349.

A076836 Number of inequivalent indecomposable binary linear [n,k] codes with no column of zeros of any dimension k <= n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 30, 76, 220, 700, 2520, 10503, 51368, 306328, 2313432, 23069977, 314605256, 5991456377, 160321885780, 6008649072476, 313490988938680, 22641945794083024, 2254587340059129076, 308683056074116543631

Offset: 1

N. J. A. Sloane, Nov 21 2002

nonn

D. Slepian, Some further theory of group codes. Bell System Tech. J. 39 1960 1219-1252.

Index entries for sequences related to binary linear codes

Row sums of A034254.

More terms from Vladeta Jovovic, Nov 27 2008

A034350 Number of indecomposable binary [ n,3 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 10, 18, 31, 51, 79, 121, 177, 254, 356, 490, 661, 882, 1157, 1501, 1926, 2445, 3073, 3834, 4740

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254, A034344-.

A034351 Number of indecomposable binary [ n,4 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 10, 28, 71, 165, 361, 754, 1503, 2893, 5393, 9773, 17273, 29860, 50557, 84024, 137228, 220542, 349128, 544980, 839453

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254, A034344-.

A034352 Number of indecomposable binary [ n,5 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 18, 71, 250, 809, 2484, 7240, 20341, 55322, 146237, 376725, 947555, 2328999, 5598888, 13171906, 30342861, 68481058, 151512767, 328820214

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254, A034344-.

cp's OEIS Frontend

A034358 Number of binary [ n,4 ] codes.

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A034357 Number of binary [ n,3 ] codes.

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A034359 Number of binary [ n,5 ] codes.

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A034360 Number of binary [ n,6 ] codes.

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A034361 Number of binary [ n,7 ] codes.

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A034362 Number of binary [ n,8 ] codes.

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A076836 Number of inequivalent indecomposable binary linear [n,k] codes with no column of zeros of any dimension k <= n.

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A034350 Number of indecomposable binary [ n,3 ] codes without 0 columns.

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A034351 Number of indecomposable binary [ n,4 ] codes without 0 columns.

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A034352 Number of indecomposable binary [ n,5 ] codes without 0 columns.

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