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A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).

%I A034356 #35 Mar 14 2020 17:38:48
%S A034356 1,2,1,3,3,1,4,6,4,1,5,10,10,5,1,6,16,22,16,6,1,7,23,43,43,23,7,1,8,
%T A034356 32,77,106,77,32,8,1,9,43,131,240,240,131,43,9,1,10,56,213,516,705,
%U A034356 516,213,56,10,1,11,71,333,1060,1988,1988,1060,333,71,11,1,12,89
%N A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).
%H A034356 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034356 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>. [This is a rectangular array whose lower triangle contains T(n,k).]
%H A034356 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. [We have T(n,k) = W_{nk2}; see p. 4 of the preprint.]
%H A034356 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. [We have T(n,k) = W_{nk2}; see p. 197.]
%H A034356 Petros Hadjicostas, <a href="/A034358/a034358.txt">Generating function for column k=4</a>.
%H A034356 Petros Hadjicostas, <a href="/A034356/a034356.txt">Generating function for column k=5</a>.
%H A034356 Petros Hadjicostas, <a href="/A034356/a034356_1.txt">Generating function for column k=6</a>.
%H A034356 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 A034356 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 A034356 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 A034356 Marcel 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 A034356 Marcel 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 A034356 Marcel Wild, <a href="https://doi.org/10.1137/S0895480104445538">The asymptotic number of binary codes and binary matroids</a>, SIAM J. Discrete Math. 19(3) (2005), 691-699. [This paper apparently corrects errors in previous papers.]
%H A034356 <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>
%F A034356 From _Petros Hadjicostas_, Sep 30 2019: (Start)
%F A034356 T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.
%F A034356 G.f. for column k=1: x/(1-x)^2.
%F A034356 G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).
%F A034356 G.f. for column k=3: -(x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^8).
%F A034356 G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. For some cases, see also the links above.
%F A034356 (End)
%e A034356 Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
%e A034356   1;
%e A034356   2,  1;
%e A034356   3,  3,  1;
%e A034356   4,  6,  4,   1;
%e A034356   5, 10, 10,   5,  1;
%e A034356   6, 16, 22,  16,  6,  1;
%e A034356   7, 23, 43,  43, 23,  7, 1;
%e A034356   8, 32, 77, 106, 77, 32, 8, 1;
%e A034356   ...
%o A034356 (Sage) # Fripertinger's method to find the g.f. of column k >= 2 (for small k):
%o A034356 def A034356col(k, length):
%o A034356     R = PowerSeriesRing(ZZ, 'x', default_prec=length)
%o A034356     x = R.gen().O(length)
%o A034356     G1 = PSL(k, GF(2))
%o A034356     G2 = PSL(k-1, GF(2))
%o A034356     D1 = G1.cycle_index()
%o A034356     D2 = G2.cycle_index()
%o A034356     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
%o A034356     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
%o A034356     f = (f1 - f2)/(1-x)
%o A034356     return f.list()
%o A034356 # For instance the Taylor expansion for column k = 4 gives
%o A034356 print(A034356col(4, 30)) # _Petros Hadjicostas_, Oct 07 2019
%Y A034356 This is A076831 with the k=0 column omitted.
%Y A034356 Columns include A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).
%Y A034356 Cf. A034253, A034254, A034328.
%K A034356 tabl,nonn
%O A034356 1,2
%A A034356 _N. J. A. Sloane_