A034253 Triangle read by rows: T(n,k) = number of inequivalent linear [n,k] binary codes without 0 columns (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 6, 12, 11, 5, 1, 1, 7, 21, 27, 17, 6, 1, 1, 9, 34, 63, 54, 25, 7, 1, 1, 11, 54, 134, 163, 99, 35, 8, 1, 1, 13, 82, 276, 465, 385, 170, 47, 9, 1, 1, 15, 120, 544, 1283, 1472, 847, 277, 61, 10, 1, 1, 18, 174, 1048, 3480

Offset: 1

Author

N. J. A. Sloane

Keywords

• tabl
• nonn

Comments

"A linear (n, k)-code has columns of zeros, if and only if there is some i ∈ n such that x_i = 0 for all codewords x, and so we should exclude such columns." [Fripertinger and Kerber (1995, p. 196)] - Petros Hadjicostas, Sep 30 2019

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1   1;
  1   2   1;
  1   3   3    1;
  1   4   6    4    1;
  1   6  12   11    5   1;
  1,  7, 21,  27,  17,  6,  1;
  1,  9, 34,  63,  54, 25,  7, 1;
  1, 11, 54, 134, 163, 99, 35, 8, 1;
  ...
		

Links

• Discrete algorithms at the University of Bayreuth, Symmetrica.
• Harald Fripertinger, Isometry Classes of Codes.
• Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [This is a lower triangular array whose lower triangle contains T(n,k). In the papers, the notation S_{nk2} is used.]
• 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. [Here S_{nk2} = T(n,k).]
• Petros Hadjicostas, Generating function for column k = 4. [Cf. A034345.]
• Petros Hadjicostas, Generating function for column k = 5. [Cf. A034346.]
• Petros Hadjicostas, Generating function for column k = 6. [Cf. A034347.]
• Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
• 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.

Crossrefs

Cf. A000012 (column k=1), A253186 (column k=2), A034344 (column k=3), A034345 (column k=4), A034346 (column k=5), A034347 (column k=6), A034348 (column k=7), A034349 (column k=8).

Cf. A034254.

Programs

Sage

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A034253col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = f1 - f2
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives
print(A034253col(4, 30)) # Petros Hadjicostas, Sep 30 2019

Formula

From Petros Hadjicostas, Sep 30 2019: (Start)

T(n,k=2) = floor(n/2) + floor((n^2 + 6)/12) = A253186(n).

T(n,k) = A076832(n,k) - A076832(n,k-1) for n, k >= 1, where we define A076832(n,0) := 0 for n >= 1.

G.f. for column k=2: (x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^3).

G.f. for column k=3: (x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((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).

G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. See also some of the links above.

(End)