A034340 - OEIS

%I #41 Oct 02 2019 05:11:51

%S 1,2,4,8,16,36,79,186,462,1222,3435,10397,33578,115624,419789,1585217,

%T 6130130,23928972,93161541,358570997,1356178991,5021267644,

%U 18160079508,64086748267,220608700157,740874170362,2428425908049,7773761586051,24320543815203,74418574119634,222890851257632

%N Number of binary [ n,6 ] codes of dimension <= 6 without zero columns.

%C 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

%H Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>. [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.]

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables_3.html">Tnk2: Number of the isometry classes of all binary (n,r)-codes for 1 <= r <= k without zero-columns</a>. [This is a rectangular array whose lower triangle is A076832(n,k). Here we have column k = 6.]

%H Harald Fripertinger, <a href="https://imsc.uni-graz.at/fripertinger/codes_bms.html">Enumeration of isometry classes of linear (n,k)-codes over GF(q) in SYMMETRICA</a>, 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.]

%H Harald Fripertinger, <a href="https://doi.org/10.1016/S0024-3795(96)00530-7">Cycle of indices of linear, affine, and projective groups</a>, 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 H. Fripertinger and A. Kerber, <a href="https://doi.org/10.1007/3-540-60114-7_15">Isometry classes of indecomposable linear codes</a>. 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.]

%H Petros Hadjicostas, <a href="/A034340/a034340.txt">Generating function for a(n)</a>.

%H David Slepian, <a href="https://archive.org/details/bstj39-5-1219">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H David Slepian, <a href="https://doi.org/10.1002/j.1538-7305.1960.tb03958.x">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.

%o (Sage) # Fripertinger's method to find the g.f. of column k for small k:

%o def Tcol(k, length):

%o     G = PSL(k, GF(2))

%o     D = G.cycle_index()

%o     f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)

%o     return f.taylor(x, 0, length).list()

%o # For instance the Taylor expansion for column k = 6 gives a(n):

%o print(Tcol(6, 30)) #  _Petros Hadjicostas_, Sep 30 2019

%Y Column k=6 of A076832 (starting at n=6).

%Y Cf. A034337.

%K nonn

%O 1,2

%A _N. J. A. Sloane_.

%E More terms from _Petros Hadjicostas_, Sep 30 2019