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A034337 Number of inequivalent binary [ n,3 ] codes of dimension <= 3 without zero columns. 7
1, 2, 4, 7, 11, 19, 29, 44, 66, 96, 136, 193, 265, 361, 485, 643, 841, 1093, 1401, 1782, 2248, 2811, 3487, 4301, 5263, 6403, 7745, 9315, 11141, 13266, 15714, 18534, 21768, 25461, 29663, 34439, 39835, 45926, 52780, 60469, 69071, 78684, 89382, 101276, 114468, 129066, 145186, 162967, 182523 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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 = 3. That is, a(n) = T_{n,3,2} = A076832(n,3), but we start at n = 1 rather than at n = 3.]
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 = 3.]
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 = 3.]
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 = 3.]
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 = 3.]
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.
FORMULA
G.f.: -(x^10 - x^8 + x^6 + x^5 + x^4 - x^2 + 1)*(x^4 - x^3 + x^2 - x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^7). - Petros Hadjicostas, Sep 30 2019
PROG
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 = 3 gives a(n):
print(Tcol(3, 30)) # Petros Hadjicostas, Sep 30 2019
CROSSREFS
Column k=3 of A076832 (starting at n=3).
Cf. A034253.
Sequence in context: A326596 A170804 A024622 * A083024 A003292 A007864
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms by Petros Hadjicostas, Sep 30 2019
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)