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A091325
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Triangle T(n,k) read by rows giving number of inequivalent even binary linear [n,k] codes (n >= 1, 0 <= k <= n-1).
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3
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| "Even" means that every word has even weight. Equivalently, the all-ones vector is in the dual code.
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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
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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).
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EXAMPLE
| Triangle begins
1
1 1
1 1 1
1 2 2 1
1 2 3 2 1
1 3 5 5 3 1
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PROG
| (MAGMA) P<t> := PolynomialAlgebra(Rationals()); qbinom := function(n, k) return &*[Rationals()|(1-2^(n+1-i))/(1-2^i):i in [1..k]]; end function;
(MAGMA, cont.) for n in [2..9] do G := Sym(n); refmod := PermutationModule(G, GF(2)); refmod := refmod/sub<refmod|[1:i in [1..n]]>; 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<G|c[3]>))]:c in CL|c[1] ne 1])/#G; end for;
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CROSSREFS
| Row sums give A091326.
Sequence in context: A176298 A169623 A011847 * A193596 A143974 A035463
Adjacent sequences: A091322 A091323 A091324 * A091326 A091327 A091328
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KEYWORD
| nonn,tabl,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 01 2004
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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).
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