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| Paper IPM / M / 2296 |
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| Abstract: | |||||
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A t-(v,kλ) directed design (or simply a
t-(v,k,λ)DD) is a pair (V,B), where V is a v-set
and B is a collection of (transitively) ordered k-tuples of
distinct elements of V, such that every ordered t-tuple of
distinct elements of V belongs to exactly λ elements of
B. (We say that a ttuple belongs to a k-tuple, if its
components are contained in that k-tuple as a set, and they
appear with the same order). In this paper with a linear algebraic
approach, we study the t-tuple inclusion matrices Dv1,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 semi-triangular basis for the null space of Dvt,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 semi-triangular basis for
the null space of Dt,t+1t+1.
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