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A linear algebraic approach to directed designs

Mahmoodian, E. S ; Sharif University of Technology | 2001

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  1. Type of Document: Article
  2. Publisher: 2001
  3. Abstract:
  4. A t-(v,k,λ) 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 t-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 Dt,kv, which sheds light to the existence problem for directed designs. Among 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 Dt,t+1v. 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
  5. Keywords:
  7. Source: Australasian Journal of Combinatorics ; Volume 23 , 2001 , Pages 119-134 ; 10344942 (ISSN)
  8. URL: https://ajc.maths.uq.edu.au/pdf/23/ocr-ajc-v23-p119.pdf