A034346 Number of binary [ n,5 ] codes without 0 columns.

0, 0, 0, 0, 1, 5, 17, 54, 163, 465, 1283, 3480, 9256, 24282, 62812, 160106, 401824, 992033, 2406329, 5730955, 13393760, 30709772, 69079030, 152473837, 330344629, 702839150, 1469214076, 3019246455, 6103105779, 12142291541, 23790590387, 45932253637, 87434850942, 164188881007

Offset

1,6

Links

Table of n, a(n) for n=1..34.

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=5.]

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,5,2}.]

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.

Prog

(Sage) # Fripertinger's method to find the g.f. of column k >= 2 (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 = 5 gives a(n):
print(A034253col(5, 30)) # Petros Hadjicostas, Oct 04 2019

Crossrefs

Cf. A034254, A034344, A034345, A034347, A034348, A034349, A253186.

Column k=5 of A034253 and first differences of A034359.

Sequence in context: A295163 A195689 A079363 * A055419 A027091 A183712

Adjacent sequences: A034343 A034344 A034345 * A034347 A034348 A034349

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Petros Hadjicostas, Oct 04 2019

Status

approved