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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). 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..62.

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

EXAMPLE

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; ...

MAPLE

# 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

PROG

(Sage) # 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

CROSSREFS

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.

Sequence in context: A163311 A210555 A008949 * A078925 A072506 A188236

Adjacent sequences:  A076829 A076830 A076831 * A076833 A076834 A076835

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Nov 21 2002

EXTENSIONS

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

STATUS

approved

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Last modified May 26 13:49 EDT 2020. Contains 334626 sequences. (Running on oeis4.)