On Theory and Applications of BIB Designs with Repeated Blocks

Abstract

Consider BIB designs with parameters $v, b, r, k$ and $\lambda$. Define the support of a BIB design to be the set of its distinct blocks and let the cardinality of the support be $b^\ast$. If $b^\ast < b$ then the design is said to be a BIB design with repeated blocks. Some potential applications of such designs to experimental design and controlled sampling are given. Some necessary and sufficient conditions for the existence of these designs and some algorithms for their constructions are provided. Bounds on $b^\ast$ have been obtained. A necessary and sufficient condition under which a set of blocks can be the support of a BIB design are found. A table of BIB designs with $22 \leqq b^\ast \leqq 56$ for $v = 8$ and $k = 3$ is included.