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A034341

Number of binary [ n,7 ] codes of dimension <= 7 without zero columns.

1, 2, 4, 8, 16, 36, 80, 193, 497, 1392, 4282, 14805, 57875, 258894, 1321280, 7570495, 47305333, 311742256, 2103025726, 14206632939, 94726167427, 618051904983, 3927156178649, 24243834619157, 145277300343585, 844969890205372, 4772180415241078, 26189419064610811, 139774809119967723

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OFFSET

1,2

COMMENTS

To get the g.f. of this sequence (with a constant 1), modify the SageMath program below (cf. function f). It is too complicated to write it here. - Petros Hadjicostas, Sep 30 2019

LINKS

Table of n, a(n) for n=1..29.

Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used by Harald Fripertinger to compute T_{nk2} = A076832(n,k) using the cycle index of PGL_k(2). Here k = 7. That is, a(n) = T_{n,7,2} = A076832(n,7), but we start at n = 1 rather than at n = 7.]

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 is A076832(n,k). Here we have column k = 7.]

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 k = 7.]

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} = A076832(n,k). Here 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. [The notation for A076832(n,k) is T_{nk2}. Here k = 7.]

David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.

Wikipedia, Cycle index.

Wikipedia, Projective linear group.

PROG

(SageMath) # Fripertinger's method to find the g.f. of column k for small k:

def Tcol(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 = 7 gives a(n):

print(Tcol(7, 30)) # Petros Hadjicostas, Sep 30 2019