The Number of Repeated Blocks in Balanced Ternary Designs with Block Size Three

Let \(D\) denote any balanced ternary design with block size three, index two, and \(\rho_2 = 1\) (that is, with each element occurring repeated in just one block). This paper shows that there exists such a design \(D\) on \(V\) elements containing exactly \(k\) pairs of repeated blocks if and only if \(V \equiv 0 \pmod{3}\), \(0\leq k \leq t_V = \frac{1}{6}V(V-3), \; \; k\neq t_V – 1, \text{and} (k,V)\neq(1,6)\).