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Paper IPM / M / 2296  


Abstract:  
A t(v,kλ) directed design (or simply a
t(v,k,λ)DD) is a pair (V,B), where V is a vset
and B is a collection of (transitively) ordered ktuples of
distinct elements of V, such that every ordered ttuple of
distinct elements of V belongs to exactly λ elements of
B. (We say that a ttuple belongs to a ktuple, if its
components are contained in that ktuple as a set, and they
appear with the same order). In this paper with a linear algebraic
approach, we study the ttuple inclusion matrices D^{v}_{1,k},
which sheds light to the existence problem for directed designs.
Among the results, we find the rank of this matrix in teh case of
0 ≤ t ≤ 4. Also in the case of 0 ≤ t ≤ 3, we introduce
a semitriangular basis for the null space of D^{v}_{t,t+1}. We
prove that when 0 ≤ t ≤ 4, the obvious necessary conditions
for the existence of t(v,k,λ) signed directed designs,
are also sufficient. Finally we find a semitriangular basis for
the null space of D_{t,t+1}^{t+1}.
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