A034358 Number of binary [ n,4 ] codes.

0, 0, 0, 1, 5, 16, 43, 106, 240, 516, 1060, 2108, 4064, 7641, 14036, 25253, 44560, 77245, 131658, 220883, 365027, 594674, 955649, 1515908, 2374875, 3676632, 5627587, 8520689, 12767557, 18941641, 27834607, 40530902, 58503994, 83741461, 118904892, 167534794, 234309554, 325373538, 448747606

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Harald Fripertinger, Isometry Classes of Codes.
• Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=4.]
• H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [Here a(n) = W_{n,4,2}; see p. 4 of the preprint.]
• 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) = W_{n,4,2}; see p. 197.]
• Petros Hadjicostas, Generating function for a(n).
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
• 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.
• M. Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
• M. Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
• Index entries for sequences related to binary linear codes

Crossrefs

Cf. A034253, A034254.

Column k=4 of both A034356 and A076831 (which are the same except for column k=0).

First differences give A034345.

Programs

Sage

# Fripertinger's method to find the g.f. of column k >= 2 of A076831 or A034356 (for small k):
def A076831col(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)/(1-x)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 (this sequence) gives
print(A076831col(4, 30)) # Petros Hadjicostas, Oct 07 2019

Extensions

More terms from Petros Hadjicostas, Oct 07 2019