A034338 -id:A034338 - cp's OEIS frontend

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).

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

A034343 Number of inequivalent binary linear codes of length n and any dimension k <= n containing no column of zeros.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 80, 194, 506, 1449, 4631, 17106, 74820, 404283, 2815595, 26390082, 344330452, 6365590987, 167062019455, 6182453531508, 319847262335488, 22968149462624180, 2277881694784784852

Offset: 1

N. J. A. Sloane

nonn

Comment from N. J. A. Sloane, Nov 27 2017 (Start)
Also, (by taking duals) number of inequivalent binary linear codes of length n and any dimension k <= n containing no codewords of weight 1.
It follows from the theorem on page 64 of Schwarzenberger (1980), this is also the number of Bravais types of orthogonal lattices in dimension n. (End)
Also the number of loopless binary matroids on n points.

R. L. E. Schwarzenberger, N-Dimensional Crystallography. Pitman, London, 1980, pages 64 and 65.
M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994

Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used to compute T_{nk2} using the cycle index of PGL_k(2). Here a(n) = T_{nn2}.]
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 main diagonal is a(n).]
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. Here a(n) = T_{nn2}.]
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. [The notation for A076832(n,k) is T_{nk2}. Here a(n) = A076832(n,k) = T_{nn2}.]
R. L. E. Schwarzenberger, Crystallography in spaces of arbitrary dimension, Proc. Camb. Phil. Soc., 76(1) (1974), 23-32.
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

Cf. A034337, A034338, A034339, A034340, A034341, A034342.

A diagonal of A076832.

a(n) = A076832(n,n). - Petros Hadjicostas, Sep 30 2019

cp's OEIS Frontend

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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