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A034342 Number of binary [ n,8 ] codes of dimension <= 8 without zero columns.

%I A034342 #29 Jul 08 2025 20:15:53
%S A034342 1,2,4,8,16,36,80,194,505,1439,4559,16580,70491,361339,2278637,
%T A034342 17745061,166540680,1794040168,20987476447,254684454328,3107605995993,
%U A034342 37418101305011,439995775004885,5025781692126252,55627454599068011,596148004493419480,6186335017615750870,62196701669630203157
%N A034342 Number of binary [ n,8 ] codes of dimension <= 8 without zero columns.
%C A034342 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. - _Petros Hadjicostas_, Sep 30 2019
%H A034342 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 = 8. That is, a(n) = T_{n,8,2} = A076832(n,8), but we start at n = 1 rather than at n = 8.]
%H A034342 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034342 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 = 8.]
%H A034342 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 = 8.]
%H A034342 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 = 8.]
%H A034342 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 = 8.]
%H A034342 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 A034342 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 A034342 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.
%H A034342 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.
%o A034342 (Sage) # Fripertinger's method to find the g.f. of column k for small k:
%o A034342 def Tcol(k, length):
%o A034342     G = PSL(k, GF(2))
%o A034342     D = G.cycle_index()
%o A034342     f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)
%o A034342     return f.taylor(x, 0, length).list()
%o A034342 # For instance the Taylor expansion for column k = 8 gives a(n):
%o A034342 print(Tcol(8, 30)) # _Petros Hadjicostas_, Sep 30 2019
%Y A034342 Column k=8 of A076832 (starting at n=8).
%Y A034342 Cf. A034337.
%K A034342 nonn
%O A034342 1,2
%A A034342 _N. J. A. Sloane_
%E A034342 More terms from _Petros Hadjicostas_, Sep 30 2019