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# 6.3 Definition

Let R ⊆ G be a (partial) ordinary difference set (for definition see RDS:Introduction). Let U ≤ G be a normal subgroup and C={g1,..., g|G:U|} be a system of representatives of G/U.

As in RDS:The Coset Signature we may define the coset signature of R relative to U.

Let U=g1,...,g|G:U| be an enumeration of G/U. An ``admissible ordered signature'' for U is a tuple (v1,...,v|G:U|) such that

| ∑ | vi=k
| ∑ | vi2=λ(|U|−1)+k
| ∑ j | vj vij = λ(|U|−1)
for gi ∉ U

holds where we index the vi by elements of G/U, so vi=vgi and write vij=vgigj. Observe that the third equation is a restriction on the ordering of the tuple (v1,...,v|G:U|). If v is an admissible ordered signature, then the multiset of v is an unordered signature.

Getting ordered admissible signatures from unordered ones can be done by taking all permutations of the unordered signature and verifying the above equations. Obviously, this method isn't very satisfying (nevertheless, the methods for testing unordered signatures from section RDS:The Coset Signature do this to find out if there is an ordered signature at all. Except that they stop when they find an ordered signature).

For ordinary difference sets in extensions of semidirect products of cyclic groups, ordered signatures may be calculated a lot easier (see RoederDiss for details).

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