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Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).
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%I #35 Mar 14 2020 17:38:48

%S 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 32,77,106,77,32,8,1,9,43,131,240,240,131,43,9,1,10,56,213,516,705,

%U 516,213,56,10,1,11,71,333,1060,1988,1988,1060,333,71,11,1,12,89

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

%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>. [This is a rectangular array whose lower triangle contains T(n,k).]

%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. [We have T(n,k) = W_{nk2}; 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. [We have T(n,k) = W_{nk2}; see p. 197.]

%H Petros Hadjicostas, <a href="/A034358/a034358.txt">Generating function for column k=4</a>.

%H Petros Hadjicostas, <a href="/A034356/a034356.txt">Generating function for column k=5</a>.

%H Petros Hadjicostas, <a href="/A034356/a034356_1.txt">Generating function for column k=6</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 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 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 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 <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>

%F From _Petros Hadjicostas_, Sep 30 2019: (Start)

%F T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.

%F G.f. for column k=1: x/(1-x)^2.

%F G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).

%F 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 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 (End)

%e Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:

%e 1;

%e 2, 1;

%e 3, 3, 1;

%e 4, 6, 4, 1;

%e 5, 10, 10, 5, 1;

%e 6, 16, 22, 16, 6, 1;

%e 7, 23, 43, 43, 23, 7, 1;

%e 8, 32, 77, 106, 77, 32, 8, 1;

%e ...

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

%o def A034356col(k, length):

%o R = PowerSeriesRing(ZZ, 'x', default_prec=length)

%o x = R.gen().O(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.list()

%o # For instance the Taylor expansion for column k = 4 gives

%o print(A034356col(4, 30)) # _Petros Hadjicostas_, Oct 07 2019

%Y This is A076831 with the k=0 column omitted.

%Y 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 Cf. A034253, A034254, A034328.

%K tabl,nonn

%O 1,2

%A _N. J. A. Sloane_