A034343 Number of inequivalent binary linear codes of length n and any dimension k <= n containing no column of zeros.

1, 2, 4, 8, 16, 36, 80, 194, 506, 1449, 4631, 17106, 74820, 404283, 2815595, 26390082, 344330452, 6365590987, 167062019455, 6182453531508, 319847262335488, 22968149462624180, 2277881694784784852

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Comment from N. J. A. Sloane, Nov 27 2017 (Start)

Also, (by taking duals) number of inequivalent binary linear codes of length n and any dimension k <= n containing no codewords of weight 1.

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)

Also the number of loopless binary matroids on n points.

References

• R. L. E. Schwarzenberger, N-Dimensional Crystallography. Pitman, London, 1980, pages 64 and 65.
• M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994

Links

• Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used to compute T_{nk2} using the cycle index of PGL_k(2). Here a(n) = T_{nn2}.]
• Harald Fripertinger, Isometry Classes of Codes.
• Harald Fripertinger, Tnk2: Number of the isometry classes of all binary (n,r)-codes for 1 <= r <= k without zero-columns. [This is a rectangular array whose main diagonal is a(n).]
• Harald Fripertinger, Enumeration of isometry classes of linear (n,k)-codes over GF(q) in SYMMETRICA, 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}.]
• Harald Fripertinger, Cycle of indices of linear, affine, and projective groups, Linear Algebra and its Applications 263 (1997), 133-156. [See p. 152 for the computation of T_{nk2}.]
• 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. [The notation for A076832(n,k) is T_{nk2}. Here a(n) = A076832(n,k) = T_{nn2}.]
• R. L. E. Schwarzenberger, Crystallography in spaces of arbitrary dimension, Proc. Camb. Phil. Soc., 76(1) (1974), 23-32.
• 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.
• Wikipedia, Cycle index.
• Wikipedia, Projective linear group.
• Index entries for sequences related to binary linear codes

Crossrefs

Cf. A034337, A034338, A034339, A034340, A034341, A034342.

A diagonal of A076832.

Formula

a(n) = A076832(n,n). - Petros Hadjicostas, Sep 30 2019