A034347 Number of binary [ n,6 ] codes without 0 columns.
0, 0, 0, 0, 0, 1, 6, 25, 99, 385, 1472, 5676, 22101, 87404, 350097, 1413251, 5708158, 22903161, 90699398, 352749035, 1342638839, 4990325414, 18090636016, 63933709870, 220277491298, 740170023052, 2426954735273, 7770739437179, 24314436451415, 74406425640743, 222867051758565, 653898059035166
Offset: 1
Keywords
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. [See column k=6.]
- 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 a(n) = S_{n,6,2}.]
- Petros Hadjicostas, Generating function for a(n).
- 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
Programs
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SageMath
# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (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 = 6 (this sequence) gives print(A034253col(6, 30)) # Petros Hadjicostas, Oct 05 2019
Extensions
More terms from Petros Hadjicostas, Oct 05 2019