On Ternary Designs with a Specified Number of Blocks with Repeated Elements

The blocks of a balanced ternary design, \(\mathrm{BTD}(V, B; p_1, p_2, R; K, \Lambda)\), can be partitioned into two sets: the \(b_1\) blocks that each contain no repeated elements, and the \(b_2 = B – b_1\) blocks containing repeated elements. In this note, we address, and answer in some particular cases, the following question. For which partitions of the integer \(B\) as \(b_1 + b_2\) does there exist a \(\mathrm{BTD}(V, B; p_1, p_2, R; K, \Lambda)\)?