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A034338
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Number of binary [ n,4 ] codes of dimension <= 4 without zero columns.
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3
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1, 2, 4, 8, 15, 30, 56, 107, 200, 372, 680, 1241, 2221, 3938, 6880, 11860, 20148, 33778, 55814, 91007, 146392, 232458, 364462, 564560, 864230, 1308160, 1958700, 2902417, 4258009, 6187350, 8908680, 12714829, 17994860, 25262928, 35193094, 48664341, 66814595, 91109910, 123426848, 166156091
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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. See the link below. - 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 = 4. That is, a(n) = T_{n,4,2} = A076832(n,4), but we start at n = 1 rather than at n = 4.]
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 = 4.]
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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 = 4 gives a(n):
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CROSSREFS
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Column k=4 of A076832 (starting at n=4).
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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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