Abstract
Let \(K_{1(3)} = \{k : k \equiv 1 ({\rm mod}\,3)\}\) . For w ∈ K 1(3), a \((v, K_{1(3)} \cup \{w^*\})\) -PBD is a pairwise balanced design on v points with block size from the set K 1(3) in which there is at least one block of size w. In this paper, we investigate the existence problem for (v, K 1(3) ∪ {w *})-PBDs and give a complete solution to this problem. As its applications, we solve completely the embedding problem for directed designs DB(4,1;u)s. In addition, we also apply our (v, K 1(3) ∪ {w *})-PBDs to do embeddings for near resolvable triple systems and nested Steiner triple systems and give unified and simple new proofs of two known theorems. Some new 4-GDDs are constructed as well.
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Communicated by C.J. Colbourn.
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Wang, J., Shen, H. Existence of \((v, K_{1(3)}\cup\{{w}^*\})\) -PBDs and its applications. Des. Codes Cryptogr. 46, 1–16 (2008). https://doi.org/10.1007/s10623-007-9122-1
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DOI: https://doi.org/10.1007/s10623-007-9122-1