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Paper   IPM / M / 2296
School of Mathematics
  Title:   A linear algebraic approach to directed designs
1.  E. S. Mahmoodian
2.  N. Soltankhah
  Status:   Published
  Journal: Australas. J. Combin.
  Vol.:  23
  Year:  2001
  Pages:   119-134
  Supported by:  IPM
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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