A034348 Number of binary [ n,7 ] codes without 0 columns.
0, 0, 0, 0, 0, 0, 1, 7, 35, 170, 847, 4408, 24297, 143270, 901491, 5985278, 41175203, 287813284, 2009864185, 13848061942, 93369988436, 613030637339, 3908996099141, 24179747870890, 145056691643428, 844229016035010, 4769751989333029, 26181645303024760, 139750488576152520
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=7.]
- 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,7,2}.]
- 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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Sage
# 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 = 7 (this sequence) gives print(A034253col(7, 30)) #
Extensions
More terms from Petros Hadjicostas, Oct 05 2019
Comments