A034344 -id:A034344 - cp's OEIS frontend

A034253 Triangle read by rows: T(n,k) = number of inequivalent linear [n,k] binary codes without 0 columns (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 6, 12, 11, 5, 1, 1, 7, 21, 27, 17, 6, 1, 1, 9, 34, 63, 54, 25, 7, 1, 1, 11, 54, 134, 163, 99, 35, 8, 1, 1, 13, 82, 276, 465, 385, 170, 47, 9, 1, 1, 15, 120, 544, 1283, 1472, 847, 277, 61, 10, 1, 1, 18, 174, 1048, 3480

Offset: 1

N. J. A. Sloane

"A linear (n, k)-code has columns of zeros, if and only if there is some i ∈ n such that x_i = 0 for all codewords x, and so we should exclude such columns." [Fripertinger and Kerber (1995, p. 196)] - Petros Hadjicostas, Sep 30 2019

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1   1;
  1   2   1;
  1   3   3    1;
  1   4   6    4    1;
  1   6  12   11    5   1;
  1,  7, 21,  27,  17,  6,  1;
  1,  9, 34,  63,  54, 25,  7, 1;
  1, 11, 54, 134, 163, 99, 35, 8, 1;
  ...

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. [This is a lower triangular array whose lower triangle contains T(n,k). In the papers, the notation S_{nk2} is used.]
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 S_{nk2} = T(n,k).]
Petros Hadjicostas, Generating function for column k = 4. [Cf. A034345.]
Petros Hadjicostas, Generating function for column k = 5. [Cf. A034346.]
Petros Hadjicostas, Generating function for column k = 6. [Cf. A034347.]
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. A000012 (column k=1), A253186 (column k=2), A034344 (column k=3), A034345 (column k=4), A034346 (column k=5), A034347 (column k=6), A034348 (column k=7), A034349 (column k=8).

Cf. A034254.

# 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 = 4 gives
print(A034253col(4, 30)) # Petros Hadjicostas, Sep 30 2019

From Petros Hadjicostas, Sep 30 2019: (Start)
T(n,k=2) = floor(n/2) + floor((n^2 + 6)/12) = A253186(n).
T(n,k) = A076832(n,k) - A076832(n,k-1) for n, k >= 1, where we define A076832(n,0) := 0 for n >= 1.
G.f. for column k=2: (x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^3).
G.f. for column k=3: (x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^7).
G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. See also some of the links above.
(End)

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

Original entry on oeis.org

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

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)).

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

A034253 Triangle read by rows: T(n,k) = number of inequivalent linear [n,k] binary codes without 0 columns (n >= 1, 1 <= k <= n).

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

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Sage

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

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Sage

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

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SageMath

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

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Sage

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

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Sage

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A034357 Number of binary [ n,3 ] codes.

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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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References