A034349 Number of binary [ n,8 ] codes without 0 columns.

0, 0, 0, 0, 0, 0, 0, 1, 8, 47, 277, 1775, 12616, 102445, 957357, 10174566, 119235347, 1482297912, 18884450721, 240477821389, 3012879828566, 36800049400028, 436068618826236, 5001537857507095, 55482177298724426, 595303034603214108, 6181562837200509792, 62170512250565592346

Offset

1,9

Comments

To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 07 2019

Links

Table of n, a(n) for n=1..28.

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=8.]

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,8,2}.]

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.

Prog

(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 = 8 (current sequence) gives
print(A034253col(8, 30)) # Petros Hadjicostas, Oct 07 2019

Crossrefs

Cf. A034254, A034344, A034345, A034346, A034347, A034348, A253186.

Column k=8 of A034253 and first differences of A034362.

Sequence in context: A014524 A098891 A054488 * A296797 A024108 A247726

Adjacent sequences: A034346 A034347 A034348 * A034350 A034351 A034352

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from Petros Hadjicostas, Oct 07 2019

Status

approved