|
|
A034337
|
|
Number of inequivalent binary [ n,3 ] codes of dimension <= 3 without zero columns.
|
|
7
|
|
|
1, 2, 4, 7, 11, 19, 29, 44, 66, 96, 136, 193, 265, 361, 485, 643, 841, 1093, 1401, 1782, 2248, 2811, 3487, 4301, 5263, 6403, 7745, 9315, 11141, 13266, 15714, 18534, 21768, 25461, 29663, 34439, 39835, 45926, 52780, 60469, 69071, 78684, 89382, 101276, 114468, 129066, 145186, 162967, 182523
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used by Harald Fripertinger to compute T_{nk2} = A076832(n,k) using the cycle index of PGL_k(2). Here k = 3. That is, a(n) = T_{n,3,2} = A076832(n,3), but we start at n = 1 rather than at n = 3.]
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 k = 3.]
|
|
FORMULA
|
G.f.: -(x^10 - x^8 + x^6 + x^5 + x^4 - x^2 + 1)*(x^4 - x^3 + x^2 - x + 1)/((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). - Petros Hadjicostas, Sep 30 2019
|
|
PROG
|
def Tcol(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 = 3 gives a(n):
|
|
CROSSREFS
|
Column k=3 of A076832 (starting at n=3).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|