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A034337 Number of inequivalent binary [ n,3 ] codes of dimension <= 3 without zero columns.

%I A034337 #27 Jul 08 2025 20:15:18
%S A034337 1,2,4,7,11,19,29,44,66,96,136,193,265,361,485,643,841,1093,1401,1782,
%T A034337 2248,2811,3487,4301,5263,6403,7745,9315,11141,13266,15714,18534,
%U A034337 21768,25461,29663,34439,39835,45926,52780,60469,69071,78684,89382,101276,114468,129066,145186,162967,182523
%N A034337 Number of inequivalent binary [ n,3 ] codes of dimension <= 3 without zero columns.
%H A034337 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 = 3. That is, a(n) = T_{n,3,2} = A076832(n,3), but we start at n = 1 rather than at n = 3.]
%H A034337 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034337 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 = 3.]
%H A034337 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 = 3.]
%H A034337 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 = 3.]
%H A034337 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 = 3.]
%H A034337 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 A034337 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 A034337 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.
%H A034337 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.
%F A034337 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
%o A034337 (SageMath)
%o A034337 def Tcol(k, length):
%o A034337     G = PSL(k, GF(2))
%o A034337     D = G.cycle_index()
%o A034337     f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)
%o A034337     return f.taylor(x, 0, length).list()
%o A034337 # For instance the Taylor expansion for column k = 3 gives a(n):
%o A034337 print(Tcol(3, 30)) # _Petros Hadjicostas_, Sep 30 2019
%Y A034337 Column k=3 of A076832 (starting at n=3).
%Y A034337 Cf. A034253.
%K A034337 nonn
%O A034337 1,2
%A A034337 _N. J. A. Sloane_
%E A034337 More terms by _Petros Hadjicostas_, Sep 30 2019