A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 16, 22, 16, 6, 1, 7, 23, 43, 43, 23, 7, 1, 8, 32, 77, 106, 77, 32, 8, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 11, 71, 333, 1060, 1988, 1988, 1060, 333, 71, 11, 1, 12, 89

Offset: 1

Author

N. J. A. Sloane

Keywords

• tabl
• nonn

Examples

			Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  4,   1;
  5, 10, 10,   5,  1;
  6, 16, 22,  16,  6,  1;
  7, 23, 43,  43, 23,  7, 1;
  8, 32, 77, 106, 77, 32, 8, 1;
  ...
		

Links

• Harald Fripertinger, Isometry Classes of Codes.
• Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [This is a rectangular array whose lower triangle contains T(n,k).]
• H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have T(n,k) = W_{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) = W_{nk2}; see p. 197.]
• Petros Hadjicostas, Generating function for column k=4.
• Petros Hadjicostas, Generating function for column k=5.
• Petros Hadjicostas, Generating function for column k=6.
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
• 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.
• Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
• Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
• Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19(3) (2005), 691-699. [This paper apparently corrects errors in previous papers.]
• Index entries for sequences related to binary linear codes

Crossrefs

This is A076831 with the k=0 column omitted.

Columns include A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).

Cf. A034253, A034254, A034328.

Programs

Sage

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A034356col(k, length):
    R = PowerSeriesRing(ZZ, 'x', default_prec=length)
    x = R.gen().O(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)/(1-x)
    return f.list()
# For instance the Taylor expansion for column k = 4 gives
print(A034356col(4, 30)) # Petros Hadjicostas, Oct 07 2019

Formula

From Petros Hadjicostas, Sep 30 2019: (Start)

T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.

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

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

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)^8).

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

(End)