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 suffcient with one possible exception for (v,λ) = (39,9).

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 A. 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 Av（v- 1） - 0 mod （c^2u（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）.