A076831 -id:A076831 - cp's OEIS frontend

A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 16, 22, 16, 6, 1, 7, 23, 43, 43, 23, 7, 1, 8, 32, 77, 106, 77, 32, 8, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 11, 71, 333, 1060, 1988, 1988, 1060, 333, 71, 11, 1, 12, 89

Offset: 1

N. J. A. Sloane

			Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  4,   1;
  5, 10, 10,   5,  1;
  6, 16, 22,  16,  6,  1;
  7, 23, 43,  43, 23,  7, 1;
  8, 32, 77, 106, 77, 32, 8, 1;
  ...

Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [This is a rectangular array whose lower triangle contains T(n,k).]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have T(n,k) = W_{nk2}; 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 T(n,k) = W_{nk2}; see p. 197.]
Petros Hadjicostas, Generating function for column k=4.
Petros Hadjicostas, Generating function for column k=5.
Petros Hadjicostas, Generating function for column k=6.
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.
Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19(3) (2005), 691-699. [This paper apparently corrects errors in previous papers.]
Index entries for sequences related to binary linear codes

This is A076831 with the k=0 column omitted.
Columns include A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).
Cf. A034253, A034254, A034328.

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A034356col(k, length):
    R = PowerSeriesRing(ZZ, 'x', default_prec=length)
    x = R.gen().O(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.list()
# For instance the Taylor expansion for column k = 4 gives
print(A034356col(4, 30)) # Petros Hadjicostas, Oct 07 2019

From Petros Hadjicostas, Sep 30 2019: (Start)
T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.
G.f. for column k=1: x/(1-x)^2.
G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).
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)^8).
G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. For some cases, see also the links above.
(End)

A034198 Number of binary codes (not necessarily linear) of length n with 3 words.

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 23, 32, 43, 56, 71, 89, 109, 132, 158, 187, 219, 255, 294, 337, 384, 435, 490, 550, 614, 683, 757, 836, 920, 1010, 1105, 1206, 1313, 1426, 1545, 1671, 1803, 1942, 2088, 2241, 2401, 2569, 2744, 2927, 3118, 3317, 3524, 3740

Offset: 1

N. J. A. Sloane

nonn

Number of distinct triangles on vertices of n-dimensional cube.
Also, a(n) is the number of orbits of C_2^2 subgroups of C_2^n under automorphisms of C_2^n.
Also, a(n) is the number of faithful representations of C_2^2 of dimension n up to equivalence by automorphisms of (C_2^2).
Also, a([n/2]) is equal to the number of partitions mu such that there exists a C_2^2 subgroup G of S_n such that the i^th largest (nontrivial) product of 2-cycles in G consists of mu_i 2-cycles (see below example). - John M. Campbell, Jan 22 2016

			Let t denote the trivial representation and u_1, u_2, u_3 the three nontrivial irreducible representations of C_2^2 (so the u_i are all equivalent up to automorphisms of C_2^2). Then the a(4) = 6 faithful representations of dimension 4 are:
  2t+u_1+u_2; t+2u_1+u_2; t+u_1+u_2+u_3;
  3u_1+u_2;   2u_1+2u_2;  2u_1+u_2+u_3.
From _John M. Campbell_, Jan 22 2016: (Start)
Letting n=8, there are a([n/2])=a(4)=6 partitions mu such that there exists a Klein four-subgroup G of S_n such that the i^th largest (nontrivial) product of 2-cycles in G consists of mu_i 2-cycles, as indicated below:
{2, 1, 1} <-> {(12)(34), (12), (34), id}
{3, 2, 1} <-> {(12)(34)(56), (34)(56), (12), id}
{2, 2, 2} <-> {(12)(34), (34)(56), (56)(12), id}
{4, 3, 1} <-> {(12)(34)(56)(78), (34)(56)(78), (12), id}
{4, 2, 2} <-> {(12)(34)(56)(78), (56)(78), (12)(34), id}
{3, 3, 2} <-> {(12)(34)(56), (34)(56)(78), (12)(78), id}
(End)

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.
J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
H. Fripertinger, Isometry Classes of Codes
H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619-630.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A034188.

Column k=2 of both A034356 and A076831 (which are the same except for column k=0).

[Floor(n*(2*n^2+21*n-6)/72): n in [1..50]]; // Vincenzo Librandi, Sep 18 2016

A034198 := n -> iquo(n*(2*n^2+21*n-6), 72):
seq(A034198(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[n (2n^2+21*n-6)/72],{n,50}] (* Harvey P. Dale, Dec 25 2011 *)
LinearRecurrence[ {2,0,-1,-1,0,2,-1},{0,1,3,6,10,16,23},50] (* Harvey P. Dale, Dec 25 2011 *)

a(n) = floor(n*(2*n^2 + 21*n - 6)/72).
G.f.: (-x^5 + x^3 + x^2)/((1 - x)^2*(1 - x^2)*(1 - x^3)) = 1/((1 - x)^2*(1 - x^2)*(1 - x^3)) - 1/(1 - x)^2.
a(1) = 0, a(2) = 1, a(3) = 3, a(4) = 6, a(5) = 10, a(6) = 16, a(7) = 23, and a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7) for n >= 8. [Harvey P. Dale, Dec 25 2011]
From Irena Swanson, Feb 11 2024: (Start)
The roots of the characteristic polynomial corresponding to the above recurrence are 1, 1, 1, 1, -1, -1/2 - sqrt(-3)/2 and -1/2 + sqrt(-3)/2. The corresponding closed form is:
a(n) = -25/144 - n/12 + 7n^2/24 + n^3/36 + (-1)^n/16 + (1/18 + sqrt(-3)/54)(-1/2 - sqrt(-3)/2)^n + (1/18 - sqrt(-3)/54)(-1/2 + sqrt(-3)/2)^n for n >= 1. (End)

Additional comments from Max Alekseyev, Jul 09 2006

Additional comments from Andrew Rupinski, Jan 20 2010

A076766 Number of inequivalent binary linear codes of length n. Also the number of nonisomorphic binary matroids on an n-set.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 68, 148, 342, 848, 2297, 6928, 24034, 98854, 503137, 3318732, 29708814, 374039266, 6739630253, 173801649708, 6356255181216, 326203517516704, 23294352980140884, 2301176047764925736, 313285408199180770635, 58638266023262502962716

Offset: 0

Marcel Wild (mwild(AT)sun.ac.za), Nov 14 2002

			a(2)=4 because there are four inequivalent linear binary 2-codes: {(0,0)}, {(0,0),(1,0)}, {(0,0),(1,1)}, {(0,0),(1,0),(0,1),(1,1)}. Observe that the codes {(0,0),(1,0)} and {(0,0),(0,1)} are equivalent because one arises from the other by a permutation of coordinates.

M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint Nr. 1693, Tech. Hochschule Darmstadt, 1994.

Jayant Apte and J. M. Walsh, Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding, arXiv preprint arXiv:1605.04598 [cs.IT], 2016-2017.
Suyoung Choi and Hanchul Park, Multiplication structure of the cohomology ring of real toric spaces, arXiv:1711.04983 [math.AT], 2017.
H. Fripertinger, Isometry Classes of Codes.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
James Oxley, What is a Matroid?.
Gordon Royle and Dillon Mayhew, 9-element matroids.
D. Slepian, On the number of symmetry types of Boolean functions of n variables, Canadian J. Math. 5, (1953), 185-193.
D. Slepian, A class of binary signaling alphabets, Bell System Tech. J. 35 (1956), 203-234.
D. Slepian, Some further theory of group codes, Bell System Tech. J. 39 1960 1219-1252. (Row sums of Table II.)
Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.]
Index entries for sequences related to binary linear codes

Row sums of triangle A076831. Cf. A034328, A055545.

Edited by N. J. A. Sloane, Nov 01 2007, at the suggestion of Gordon Royle.

A076832 Triangle T(n,k), read by rows, giving the total number of inequivalent binary linear [n,i] codes with no column of zeros, summed for i <= k (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 7, 19, 30, 35, 36, 1, 8, 29, 56, 73, 79, 80, 1, 10, 44, 107, 161, 186, 193, 194, 1, 12, 66, 200, 363, 462, 497, 505, 506, 1, 14, 96, 372, 837, 1222, 1392, 1439, 1448, 1449, 1, 16, 136, 680, 1963, 3435, 4282

Offset: 1

N. J. A. Sloane, Nov 21 2002

From Petros Hadjicostas, Sep 30 2019: (Start)
It seems that Harald Fripertinger at his website defines T(n,k) = T(n,n) for k > n (and thus he gets an orthogonal array). It seems that T(n,n) = A034343(n).
It seems that T(n,k=2) = A001399(n) for n >= 2 (with A001399(n=1) = T(1,1)); T(n,k=3) = A034337(n) for n >= 3 (with A034337(n) = T(n,n) for 1 <= n <= 2); T(n,k=4) = A034338(n) for n >= 4 (with A034338(n) = T(n,n) for 1 <= n <= 3); and so on. See the Crossrefs below for more information.
To get the g.f. of column k (starting at n = 0 with T(n=0,k) := 1 rather than at n = k), modify the Sage program below (cf. function f).
(End)

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1,  2;
  1,  3,  4;
  1,  4,  7,   8;
  1,  5, 11,  15,  16;
  1,  7, 19,  30,  35,  36;
  1,  8, 29,  56,  73,  79,  80;
  1, 10, 44, 107, 161, 186, 193, 194; ...

Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used to compute T_{nk2} using the cycle index of PGL_k(2).]
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Tnk2: Number of the isometry classes of all binary (n,r)-codes for 1 <= r <= k without zero-columns. [This is a rectangular array whose lower triangle contains T(n,k).]
Harald Fripertinger, Enumeration of isometry classes of linear (n,k)-codes over GF(q) in SYMMETRICA, Bayreuther Mathematische Schriften 49 (1995), 215-223. [See pp. 216-218. A C-program is given for calculating T_{nk2} in Symmetrica.]
Harald Fripertinger, Cycle of indices of linear, affine, and projective groups, Linear Algebra and its Applications 263 (1997), 133-156. [See p. 152 for the computation of T_{nk2}.]
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. [Apparently, the notation for T(n,k) is T_{nk2}.]
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.
Index entries for sequences related to binary linear codes

Columns give truncated versions of A001399 (k = 2), A034337 (k = 3), A034338 (k = 4), A034339 (k = 5), A034340 (k = 6), A034341 (k = 7), A034342 (k = 8), and A034343 (? main diagonal).

Cf. A034253, A076831.

# We illustrate how to get a g.f. for column k in Maple when k is small.
with(GroupTheory);
G := ProjectiveGeneralLinearGroup(4, 2);
GroupOrder(G);
# We get that the order is 20160.
f:=CycleIndexPolynomial(G, [x||(1..20160)]);
# We get
# 1/20160*x1^15 + 1/192*x1^7*x2^4 + 1/96*x1^3*x2^6 + 1/16*x1^3*x2^2*x4^2 +
# 1/18*x1^3*x3^4 + 1/6*x1*x2*x3^2*x6 + 1/8*x1*x2*x4^3 + 1/180*x3^5 + 2/7*x1*x7^2 +
# 1/12*x3*x6^2 + 1/15*x5^3 + 2/15*x15
# The only dummy variables that appear are x1, x2, x3, x4, x5, x6, x7, and x15.
g:=subs(x1 = 1/(1 - y), subs(x2 = 1/(-y^2 + 1), subs(x3 = 1/(-y^3 + 1), subs(x4 = 1/(-y^4 + 1), subs(x5 = 1/(-y^5 + 1), subs(x6 = 1/(-y^6 + 1), subs(x7 = 1/(-y^7 + 1), subs(x15 = 1/(-y^15 + 1), f))))))));
# Then we take a Taylor expansion of the above g.f.
taylor(g,y=0,50);
# We get a Taylor expansion for column k = 4 (i.e., A034338).
# Petros Hadjicostas, Sep 30 2019

# Fripertinger's method to find the g.f. of column k for small k:
def A076832col(k, length):
    G = PSL(k, GF(2))
    D = G.cycle_index()
    f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives A034338:
print(A076832col(4, 30)) # Petros Hadjicostas, Sep 30 2019

Revised by N. J. A. Sloane, Mar 01 2004

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.

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.

cp's OEIS Frontend

A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).

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Sage

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A034198 Number of binary codes (not necessarily linear) of length n with 3 words.

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Magma

Maple

Mathematica

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A076766 Number of inequivalent binary linear codes of length n. Also the number of nonisomorphic binary matroids on an n-set.

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A076832 Triangle T(n,k), read by rows, giving the total number of inequivalent binary linear [n,i] codes with no column of zeros, summed for i <= k (n >= 1, 1 <= k <= n).

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Maple

Sage

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A034358 Number of binary [ n,4 ] codes.

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Sage

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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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