The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B( s n , ts m , λ t ( ts m  − 1) ( s n‐m  − 1)/[2( s m  − 1)]) for any positive integer λ, such that s n ( s n  − 1) λ ≡ 0 (mod s m ( s m  − 1) and for any positive integer t with 2 ≤  t  ≤  s n‐m , where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v  = 4 k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007