The Trade-Off Method in the Construction of BIB Designs with Variable Support Sizes

Abstract

A balanced incomplete block (BIB) design with $b$ blocks is said to have the support size $b^\ast$ when exactly $b^\ast$ of the $b$ blocks are distinct. BIB designs with $b^\ast < b$ have interesting applications in design of experiments and controlled sampling as explained in details in Foody and Hedayat (1977) and Wynn (1977). A method called "trade-off" is introduced for the construction of BIB designs with repeated blocks. This method is utilized to study BIB designs with arbitrary $v$ treatments in blocks of size $k = 3$ in general and with $v = 7$ and $k = 3$ in particular. It is shown that BIB designs with $v = 7, k = 3$, any $b$, and any $b^\ast$ exist if and only if (i) $b$ is divisible by 7, (ii) $7 \leqslant b^\ast \leqslant \min(b, 35)$, (iii) $b^\ast \neq 8, 9, 10, 12$, or $16$, (iv) $(b, b^\ast) \neq (28, 24), (28, 27), (35, 30), (35, 32), (35, 33), (35, 34)$ or (42, 34).