A034254 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).

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, 71, 71, 31, 7, 1, 1, 8, 51, 165, 250, 165, 51, 8, 1, 1, 10, 79, 361, 809, 809, 361, 79, 10, 1, 1, 12, 121, 754, 2484, 3759, 2484, 754, 121, 12, 1, 1, 14, 177, 1503, 7240, 16749, 16749, 7240, 1503, 177, 14, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• tabl
• nonn

Comments

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

Examples

			Triangle T(n,k) (with rows n >= 2 and columns k >= 1) begins as follows:
  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,  71,  71,  31,  7, 1;
  1, 8, 51, 165, 250, 165, 51, 8, 1;
  ...
		

Links

• Discrete algorithms at the University of Bayreuth, Symmetrica.
• Harald Fripertinger, Isometry Classes of Codes.
• Harald Fripertinger, Rnk2: Number of the isometry classes of all binary indecomposable (n,k)-codes without zero columns. [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).]
• Harald Fripertinger, Enumeration of isometry-classes of linear (n,k)-codes over GF(q) in SYMMETRICA, 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. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have T(n,k) = R_{nk2}; see p. 4 of the preprint.]
• H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. 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.]
• David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
• David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
• Index entries for sequences related to binary linear codes

Crossrefs

Cf. A076836 (row sums), A034253.

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).

Extensions

More terms from Petros Hadjicostas, Oct 07 2019