Balanced Incomplete Block Designs and Tactical Configurations

Abstract

A balanced incomplete block design (BIB design) is an arrangement of $v$ varieties of treatments in $b$ blocks of $k$ distinct varieties each, so that each variety is contained in $r$ blocks and every pair of varieties is contained in $\lambda$ blocks. Various methods of constructing such designs are discussed in [2], and certain designs are listed in [3], [4], [5], [7], [14]. If $v = b$, the design is said to be symmetric; the impossibility of certain symmetric designs was proved in [10]. Although in [8] certain tactical configurations are discussed, it seems that the relationship between BIB designs and tactical configurations, and in particular, the Steiner system, has been overlooked. It is the purpose of this note to point out this relationship and to discuss the properties of designs arising from such configurations.