A034253 -id:A034253 - cp's OEIS frontend

A034345 Number of binary [ n,4 ] codes without 0 columns.

0, 0, 0, 1, 4, 11, 27, 63, 134, 276, 544, 1048, 1956, 3577, 6395, 11217, 19307, 32685, 54413, 89225, 144144, 229647, 360975, 560259, 858967, 1301757, 1950955, 2893102, 4246868, 6174084, 8892966, 12696295, 17973092, 25237467, 35163431, 48629902, 66774760, 91063984

Offset: 1

N. J. A. Sloane

nonn

"We say that the sequence (a_n) is quasi-polynomial in n if there exist polynomials P_0, ..., P_{s-1} and an integer n_0 such that, for all n >= n_0, a_n = P_i(n) where i == n (mod s)." [This is from the abstract of Lisonek (2007), and he states that the condition "n >= n_0" makes his definition broader than the one in Stanley's book. From Section 5 of his paper, we conclude that (a(n): n >= 1) is a quasi-polynomial in n.] - Petros Hadjicostas, Oct 02 2019

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=4.]
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) = S_{n,4,2}.]
Petros Hadjicostas, Generating function for a(n).
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
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.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034346, A034347, A034348, A034349, A253186.

Column k=4 of A034253 and first differences of A034358.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(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
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 (this sequence) gives
print(A034253col(4, 30)) #

More terms by Petros Hadjicostas, Oct 02 2019

A034346 Number of binary [ n,5 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 17, 54, 163, 465, 1283, 3480, 9256, 24282, 62812, 160106, 401824, 992033, 2406329, 5730955, 13393760, 30709772, 69079030, 152473837, 330344629, 702839150, 1469214076, 3019246455, 6103105779, 12142291541, 23790590387, 45932253637, 87434850942, 164188881007

Offset: 1

N. J. A. Sloane

nonn

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=5.]
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) = S_{n,5,2}.]
Petros Hadjicostas, Generating function for a(n).
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
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.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034347, A034348, A034349, A253186.

Column k=5 of A034253 and first differences of A034359.

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A034253col(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
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 5 gives a(n):
print(A034253col(5, 30)) # Petros Hadjicostas, Oct 04 2019

More terms from Petros Hadjicostas, Oct 04 2019

A034347 Number of binary [ n,6 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 25, 99, 385, 1472, 5676, 22101, 87404, 350097, 1413251, 5708158, 22903161, 90699398, 352749035, 1342638839, 4990325414, 18090636016, 63933709870, 220277491298, 740170023052, 2426954735273, 7770739437179, 24314436451415, 74406425640743, 222867051758565, 653898059035166

Offset: 1

N. J. A. Sloane

nonn

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=6.]
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) = S_{n,6,2}.]
Petros Hadjicostas, Generating function for a(n).
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
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.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034346, A034348, A034349, A253186.
First differences of A034360.
Column k = 6 of A034253.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(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
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 6 (this sequence) gives
print(A034253col(6, 30)) # Petros Hadjicostas, Oct 05 2019

More terms from Petros Hadjicostas, Oct 05 2019

A034348 Number of binary [ n,7 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 7, 35, 170, 847, 4408, 24297, 143270, 901491, 5985278, 41175203, 287813284, 2009864185, 13848061942, 93369988436, 613030637339, 3908996099141, 24179747870890, 145056691643428, 844229016035010, 4769751989333029, 26181645303024760, 139750488576152520

Offset: 1

N. J. A. Sloane

nonn

To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 05 2019

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=7.]
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) = S_{n,7,2}.]
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
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.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034346, A034347, A034349, A253186.

Column k=7 of A034253 and first differences of A034361.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(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
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 7 (this sequence) gives
print(A034253col(7, 30)) #

More terms from Petros Hadjicostas, Oct 05 2019

A034349 Number of binary [ n,8 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 8, 47, 277, 1775, 12616, 102445, 957357, 10174566, 119235347, 1482297912, 18884450721, 240477821389, 3012879828566, 36800049400028, 436068618826236, 5001537857507095, 55482177298724426, 595303034603214108, 6181562837200509792, 62170512250565592346

Offset: 1

N. J. A. Sloane

nonn

To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 07 2019

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=8.]
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) = S_{n,8,2}.]
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
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.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034346, A034347, A034348, A253186.

Column k=8 of A034253 and first differences of A034362.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(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
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 8 (current sequence) gives
print(A034253col(8, 30)) # Petros Hadjicostas, Oct 07 2019

More terms from Petros Hadjicostas, Oct 07 2019

A034358 Number of binary [ n,4 ] codes.

Original entry on oeis.org

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.

cp's OEIS Frontend

A034345 Number of binary [ n,4 ] codes without 0 columns.

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A034346 Number of binary [ n,5 ] codes without 0 columns.

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A034347 Number of binary [ n,6 ] codes without 0 columns.

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A034348 Number of binary [ n,7 ] codes without 0 columns.

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A034349 Number of binary [ n,8 ] codes without 0 columns.

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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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