Results on the support of BIB designs (Q1121264)

This paper presents further results on BIB designs with repeated blocks by the first author in collaboration with two new co-authors. Let \(BIB(v,b,r,k,\lambda)/b^*)\) denote a BIB (v,b,r,k,\(\lambda)\) design with precisely \(b^*\) distinct blocks. The set of all distinct blocks is called the support of the BIB design and the number \(b^*\) is called the support size of the design. The authors present new lower bounds on \(b^*\) by using information about those blocks in the support which are repeated \(\lambda\) times in the design. They also provide the following specific results: (i) the nonexistence of \(BIB(8,56t,21t,3,6t| b^*)\) designs with \(b^*\leq 21\) (the design with support size 22 being already available; (ii) the non-existence of \(BIB(11,55t,15t,3,3t| b^*)\) designs with \(b^*\leq 24\) and the existence by actual construction of BIB(11,55,15,3,3\(| 25)\); (iii) the construction of BIB(12,44,11,3,2\(| 28)\) design, which has the minimum support size within the family of BIB(12,44t,11t,3,2t) designs.