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A034358 Number of binary [ n,4 ] codes. 4


%S 0,0,0,1,5,16,43,106,240,516,1060,2108,4064,7641,14036,25253,44560,

%T 77245,131658,220883,365027,594674,955649,1515908,2374875,3676632,

%U 5627587,8520689,12767557,18941641,27834607,40530902,58503994,83741461,118904892,167534794,234309554,325373538,448747606

%N Number of binary [ n,4 ] codes.

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables_6.html">Wnk2: Number of the isometry classes of all binary (n,k)-codes</a>. [See column k=4.]

%H H. Fripertinger and A. Kerber, <a href="https://www.researchgate.net/publication/2550138_Isometry_Classes_of_Indecomposable_Linear_Codes">Isometry classes of indecomposable linear codes</a>, preprint, 1995. [Here a(n) = W_{n,4,2}; see p. 4 of the preprint.]

%H H. Fripertinger and A. Kerber, <a href="https://doi.org/10.1007/3-540-60114-7_15">Isometry classes of indecomposable linear codes</a>. 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.]

%H Petros Hadjicostas, <a href="/A034358/a034358.txt">Generating function for a(n)</a>.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%H David Slepian, <a href="https://archive.org/details/bstj39-5-1219">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H David Slepian, <a href="https://doi.org/10.1002/j.1538-7305.1960.tb03958.x">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H M. Wild, <a href="http://dx.doi.org/10.1006/eujc.1996.0026">Consequences of the Brylawski-Lucas Theorem for binary matroids</a>, European Journal of Combinatorics 17 (1996), 309-316.

%H M. Wild, <a href="http://dx.doi.org/10.1006/ffta.1999.0273">The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids</a>, Finite Fields and their Applications 6 (2000), 192-202.

%H <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>

%o (Sage) # Fripertinger's method to find the g.f. of column k >= 2 of A076831 or A034356 (for small k):

%o def A076831col(k, length):

%o G1 = PSL(k, GF(2))

%o G2 = PSL(k-1, GF(2))

%o D1 = G1.cycle_index()

%o D2 = G2.cycle_index()

%o f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)

%o f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)

%o f = (f1 - f2)/(1-x)

%o return f.taylor(x, 0, length).list()

%o # For instance the Taylor expansion for column k = 4 (this sequence) gives

%o print(A076831col(4, 30)) # _Petros Hadjicostas_, Oct 07 2019

%Y Cf. A034253, A034254.

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

%Y First differences give A034345.

%K nonn

%O 1,5

%A _N. J. A. Sloane_.

%E More terms from _Petros Hadjicostas_, Oct 07 2019

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Last modified November 20 10:09 EST 2019. Contains 329334 sequences. (Running on oeis4.)