On the Existence of (v,3×3,λ)-Splitting Balanced Incomplete Block Design with λ between 2 to 9

Let (v, u × c, λ)-splitting BIBD denote a (v, u × c, λ)-splitting balanced incomplete block design of order v with block size u × c and index λ. The necessary conditions for the existence of a (v, u × c, λ)-splitting BIBD are v ≥ uc, λ(v-1) ≡ 0 0 mod (c(u-1)) and λv(v-1) ≡ 0 mod (c2u(u-1)). In this paper, for 2≤λ≤9 the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with one possible exception for (v,λ)=(39,9).     

Minimal Enclosing of an MPT （v,λ） in a TS （v＋l,μ）

Let MPT (v,λ) denote a maximum packing of triples of order v with index λ and TS (u,λ) denote atriple system of order u with index λ. In this paper, for v≥6, we determine necessary and sufficient conditions forenclosing an MPT (v，λ) in a TS (v l ,λ m) with one extra point and minimal increase in λ.