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%I #27 Oct 01 2019 09:09:39
%S 1,2,4,8,16,36,80,194,506,1449,4631,17106,74820,404283,2815595,
%T 26390082,344330452,6365590987,167062019455,6182453531508,
%U 319847262335488,22968149462624180,2277881694784784852
%N Number of inequivalent binary linear codes of length n and any dimension k <= n containing no column of zeros.
%C Comment from _N. J. A. Sloane_, Nov 27 2017 (Start)
%C Also, (by taking duals) number of inequivalent binary linear codes of length n and any dimension k <= n containing no codewords of weight 1.
%C It follows from the theorem on page 64 of Schwarzenberger (1980), this is also the number of Bravais types of orthogonal lattices in dimension n. (End)
%C Also the number of loopless binary matroids on n points.
%D R. L. E. Schwarzenberger, N-Dimensional Crystallography. Pitman, London, 1980, pages 64 and 65.
%D M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994
%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 to compute T_{nk2} using the cycle index of PGL_k(2). Here a(n) = T_{nn2}.]
%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 main diagonal is a(n).]
%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 a(n) = T_{nn2}.]
%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}.]
%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 a(n) = A076832(n,k) = T_{nn2}.]
%H R. L. E. Schwarzenberger, <a href="https://doi.org/10.1017/S0305004100048696">Crystallography in spaces of arbitrary dimension</a>, Proc. Camb. Phil. Soc., 76(1) (1974), 23-32.
%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>.
%H <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>
%F a(n) = A076832(n,n). - _Petros Hadjicostas_, Sep 30 2019
%Y Cf. A034337, A034338, A034339, A034340, A034341, A034342.
%Y A diagonal of A076832.
%K nonn
%O 1,2
%A _N. J. A. Sloane_.
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