A076832 Triangle T(n,k), read by rows, giving the total number of inequivalent binary linear [n,i] codes with no column of zeros, summed for i <= k (n >= 1, 1 <= k <= n).

1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 7, 19, 30, 35, 36, 1, 8, 29, 56, 73, 79, 80, 1, 10, 44, 107, 161, 186, 193, 194, 1, 12, 66, 200, 363, 462, 497, 505, 506, 1, 14, 96, 372, 837, 1222, 1392, 1439, 1448, 1449, 1, 16, 136, 680, 1963, 3435, 4282

Offset: 1

Author

N. J. A. Sloane, Nov 21 2002

Keywords

• nonn
• tabl

Comments

From Petros Hadjicostas, Sep 30 2019: (Start)

It seems that Harald Fripertinger at his website defines T(n,k) = T(n,n) for k > n (and thus he gets an orthogonal array). It seems that T(n,n) = A034343(n).

It seems that T(n,k=2) = A001399(n) for n >= 2 (with A001399(n=1) = T(1,1)); T(n,k=3) = A034337(n) for n >= 3 (with A034337(n) = T(n,n) for 1 <= n <= 2); T(n,k=4) = A034338(n) for n >= 4 (with A034338(n) = T(n,n) for 1 <= n <= 3); and so on. See the Crossrefs below for more information.

To get the g.f. of column k (starting at n = 0 with T(n=0,k) := 1 rather than at n = k), modify the Sage program below (cf. function f).

(End)

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1,  2;
  1,  3,  4;
  1,  4,  7,   8;
  1,  5, 11,  15,  16;
  1,  7, 19,  30,  35,  36;
  1,  8, 29,  56,  73,  79,  80;
  1, 10, 44, 107, 161, 186, 193, 194; ...
		

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).]
• 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 lower triangle contains T(n,k).]
• 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.]
• 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. [Apparently, the notation for T(n,k) is T_{nk2}.]
• 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

Columns give truncated versions of A001399 (k = 2), A034337 (k = 3), A034338 (k = 4), A034339 (k = 5), A034340 (k = 6), A034341 (k = 7), A034342 (k = 8), and A034343 (? main diagonal).

Cf. A034253, A076831.

Programs

Maple

# We illustrate how to get a g.f. for column k in Maple when k is small.
with(GroupTheory);
G := ProjectiveGeneralLinearGroup(4, 2);
GroupOrder(G);
# We get that the order is 20160.
f:=CycleIndexPolynomial(G, [x||(1..20160)]);
# We get
# 1/20160*x1^15 + 1/192*x1^7*x2^4 + 1/96*x1^3*x2^6 + 1/16*x1^3*x2^2*x4^2 +
# 1/18*x1^3*x3^4 + 1/6*x1*x2*x3^2*x6 + 1/8*x1*x2*x4^3 + 1/180*x3^5 + 2/7*x1*x7^2 +
# 1/12*x3*x6^2 + 1/15*x5^3 + 2/15*x15
# The only dummy variables that appear are x1, x2, x3, x4, x5, x6, x7, and x15.
g:=subs(x1 = 1/(1 - y), subs(x2 = 1/(-y^2 + 1), subs(x3 = 1/(-y^3 + 1), subs(x4 = 1/(-y^4 + 1), subs(x5 = 1/(-y^5 + 1), subs(x6 = 1/(-y^6 + 1), subs(x7 = 1/(-y^7 + 1), subs(x15 = 1/(-y^15 + 1), f))))))));
# Then we take a Taylor expansion of the above g.f.
taylor(g,y=0,50);
# We get a Taylor expansion for column k = 4 (i.e., A034338).
# Petros Hadjicostas, Sep 30 2019

Sage

# Fripertinger's method to find the g.f. of column k for small k:
def A076832col(k, length):
    G = PSL(k, GF(2))
    D = G.cycle_index()
    f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives A034338:
print(A076832col(4, 30)) # Petros Hadjicostas, Sep 30 2019

Extensions

Revised by N. J. A. Sloane, Mar 01 2004