A091325 Triangle T(n,k) read by rows giving number of inequivalent even binary linear [n,k] codes (n >= 1, 0 <= k <= n-1).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 9, 7, 3, 1, 1, 4, 10, 17, 17, 10, 4, 1, 1, 4, 13, 26, 35, 26, 13, 4, 1

Offset: 1

Author

N. J. A. Sloane, Mar 01 2004

Keywords

• nonn
• tabl
• more

Comments

"Even" means that every word has even weight. Equivalently, the all-ones vector is in the dual code.

Examples

			Triangle begins
1
1 1
1 1 1
1 2 2 1
1 2 3 2 1
1 3 5 5 3 1
		

Links

• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Index entries for sequences related to binary linear codes

Crossrefs

Row sums give A091326.

Programs

Magma

P := PolynomialAlgebra(Rationals()); qbinom := function(n,k) return &*[Rationals()|(1-2^(n+1-i))/(1-2^i):i in [1..k]]; end function;
for n in [2..9] do G := Sym(n); refmod := PermutationModule(G,GF(2)); refmod := refmod/sub; CL := ConjugacyClasses(G); acc := &+[qbinom(n-1,k)*t^k:k in [0..n-1]]; n,(acc+&+[P|c[2]*&+[t^(n-1-Dimension(s)):s in Submodules(Restriction(refmod,sub))]:c in CL|c[1] ne 1])/#G; end for;

Formula

T(n, 0) = T(n, n-1) = 1, T(n, n) = 0; T(n, 1) = floor(n/2); T(n, k) = T(n, n-k-1).

Extensions

Rows 7 - 9 computed by Eric Rains (rains(AT)caltech.edu) using MAGMA, Mar 01 2004

It would be nice even to have a continuation of the numbers for dimension 2, T(n,2).