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A034340 Number of binary [ n,6 ] codes of dimension <= 6 without zero columns. 3
1, 2, 4, 8, 16, 36, 79, 186, 462, 1222, 3435, 10397, 33578, 115624, 419789, 1585217, 6130130, 23928972, 93161541, 358570997, 1356178991, 5021267644, 18160079508, 64086748267, 220608700157, 740874170362, 2428425908049, 7773761586051, 24320543815203, 74418574119634, 222890851257632 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
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 = 6. That is, a(n) = T_{n,6,2} = A076832(n,6), but we start at n = 1 rather than at n = 6.]
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 = 6.]
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 = 6.]
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 = 6.]
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 = 6.]
Petros Hadjicostas, Generating function for a(n).
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.
PROG
(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 = 6 gives a(n):
print(Tcol(6, 30)) # Petros Hadjicostas, Sep 30 2019
CROSSREFS
Column k=6 of A076832 (starting at n=6).
Cf. A034337.
Sequence in context: A266544 A348414 A333051 * A034341 A341536 A034342
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Petros Hadjicostas, Sep 30 2019
STATUS
approved

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Last modified March 28 04:05 EDT 2024. Contains 371235 sequences. (Running on oeis4.)