|
%I #32 Oct 09 2019 02:48:26
%S 1,1,1,1,1,1,1,2,2,1,1,3,5,3,1,1,4,10,10,4,1,1,5,18,28,18,5,1,1,7,31,
%T 71,71,31,7,1,1,8,51,165,250,165,51,8,1,1,10,79,361,809,809,361,79,10,
%U 1,1,12,121,754,2484,3759,2484,754,121,12,1,1,14,177,1503,7240,16749,16749,7240,1503,177,14,1
%N Triangle read by rows giving T(n,k) = number of inequivalent indecomposable linear [ n,k ] binary codes without 0 columns (n >= 2, 1 <= k <= n).
%C Fripertinger and Kerber (1995) mention that Slepian (1960) gave a generating function scheme for computing R_{n,k,2} = T(n,k), but it is not always correct. In Theorem 3.1, they give a corrected formula, but it seems too difficult to implement it in Sage. They do provide, however, a SYMMETRICA program for its computation (see the links). - _Petros Hadjicostas_, Oct 07 2019
%H Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.
%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_5.html">Rnk2: Number of the isometry classes of all binary indecomposable (n,k)-codes without zero columns</a>. [This is a rectangular array, denoted by R_{nk2}, whose lower triangle (starting at n = 2) contains the current array T(n,k). The element R_{n=1,k=1,2} = 1 does not appear in the current array T(n,k).]
%H Harald Fripertinger, <a href="https://imsc.uni-graz.at/fripertinger/papers/art11.pdf">Enumeration of isometry-classes of linear (n,k)-codes over GF(q) in SYMMETRICA</a>, Bayreuther Mathematische Schriften 49 (1999), 215-223. [For a SYMMETRICA program for the calculation of R_{nk2} = T(n,k), see pp. 219-220.]
%H H. Fripertinger and A. Kerber, <a href="https://www.researchgate.net/publication/2550138_Isometry_Classes_of_Indecomposable_Linear_Codes">Isometry classes of indecomposable linear codes</a>, preprint, 1995. [We have T(n,k) = R_{nk2}; see p. 4 of the preprint.]
%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. [We have T(n,k) = R_{nk2}; see p. 197.]
%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 <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>
%e Triangle T(n,k) (with rows n >= 2 and columns k >= 1) begins as follows:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 3, 5, 3, 1;
%e 1, 4, 10, 10, 4, 1;
%e 1, 5, 18, 28, 18, 5, 1;
%e 1, 7, 31, 71, 71, 31, 7, 1;
%e 1, 8, 51, 165, 250, 165, 51, 8, 1;
%e ...
%Y Cf. A076836 (row sums), A034253.
%Y Columns include A000012 (k=1), A069905 (k=2), A034350 (k=3), A034351 (k=4), A034352 (k=5), A034353 (k=6), A034354 (k=7), A034355 (k=8).
%K tabl,nonn
%O 1,8
%A _N. J. A. Sloane_
%E More terms from _Petros Hadjicostas_, Oct 07 2019
|
