A linear algebraic approach to directed designs

A t-(v, k,;\) directed design (or simply a t-(v, k, ;\)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). elements of B. (We say that at-tuple 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 Dr k' which sheds light to the existence problem for directed designs. A~ong the results, we find the rank of this matrix in the case of 0 ~ t ~ 4. Also in the case of 0 ~ t ~ 3 , we introduce a semi-triangular basis for the null space of Df,t+l' We prove that when 0 :::; t :::; 4 , the obvious necessary conditions for the existence of t-( v, k, ;\) signed directed designs, are also sufficient. Finally we find a semi-triangular basis for the null space of DHI t,t+l' Australasian Journal of Combinatorics 23(2001). pp.119-134