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A034339 Number of binary [ n,5 ] codes of dimension <= 5 without zero columns. 3
1, 2, 4, 8, 16, 35, 73, 161, 363, 837, 1963, 4721, 11477, 28220, 69692, 171966, 421972, 1025811, 2462143, 5821962, 13540152, 30942230, 69443492, 153038397, 331208859, 704147310, 1471172776, 3022148872, 6107363788, 12148478891, 23799499067, 45944968466, 87452845802, 164214143935 (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

Table of n, a(n) for n=1..34.

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 = 5. That is, a(n) = T_{n,5,2} = A076832(n,5), but we start at n = 1 rather than at n = 5.]

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 = 5.]

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 = 5.]

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 = 5.]

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 = 5.]

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.

Wikipedia, Projective linear group.

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 = 5 gives a(n):

print(Tcol(5, 30)) # Petros Hadjicostas, Sep 30 2019

CROSSREFS

Column k=5 of A076832 (starting at n=5).

Cf. A034337.

Sequence in context: A336009 A180207 A013025 * A244519 A247293 A337716

Adjacent sequences: A034336 A034337 A034338 * A034340 A034341 A034342

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Petros Hadjicostas, Sep 30 2019

STATUS

approved

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Last modified February 5 09:25 EST 2023. Contains 360084 sequences. (Running on oeis4.)