Balanced ternary designs with block size four
A \(\text{BTD}(v,\rho_ 2,4,\lambda)\) is a balanced ternary design such that \(k=4\) and each element occurs two times in \(\rho_ 2\) blocks (and, of course, each pair of distinct elements occurs \(\lambda\) times in a block). A necessary existence condition for such a BTD is: if \(\lambda=2\) then \(v \geq 5 \rho_ 2+1\), if \(\lambda=3\) then \(v \geq 4 \rho_ 2 +1\), if \(\lambda \geq 4\) then \(v \geq \lceil 4 \rho_ 2/ \lambda \rceil+1\). The authors show the above conditions are sufficient when \(\rho_ 2 \leq 6\) teaching how to construe such designs by variations of known constructions (mainly Hanani's is used ;see \textit{H. Hanani} [Balanced incomplete block designs and related designs, Discrete Math. 11, 255-369 (1975; Zbl 0361.62067)]). They start from group divisible designs and frame-BTD, previously known or given in a huge table as appendix.
scientific article; zbMATH DE number 445193
scientific article