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A034342
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Number of binary [ n,8 ] codes of dimension <= 8 without zero columns.
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2
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1, 2, 4, 8, 16, 36, 80, 194, 505, 1439, 4559, 16580, 70491, 361339, 2278637, 17745061, 166540680, 1794040168, 20987476447, 254684454328, 3107605995993, 37418101305011, 439995775004885, 5025781692126252, 55627454599068011, 596148004493419480, 6186335017615750870, 62196701669630203157
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OFFSET
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1,2
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COMMENTS
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To get the g.f. of this sequence (with a constant 1), modify the Sage program below (cf. function f). It is too complicated to write it here. - Petros Hadjicostas, Sep 30 2019
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LINKS
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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 = 8. That is, a(n) = T_{n,8,2} = A076832(n,8), but we start at n = 1 rather than at n = 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. [The notation for A076832(n,k) is T_{nk2}. Here k = 8.]
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PROG
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(Sage) # 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 = 8 gives a(n):
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CROSSREFS
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Column k=8 of A076832 (starting at n=8).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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