The spectrum of optimal strong partially balanced designs with block size five
Let \(X\) be a set with \(v\) elements, and \(B\) be a collections of \(b\) subsets (called blocks) of \(X\) where each subset is of size \(k\). The pair \((X, B)\) is called a partially balanced \(t\)-design \((t\leq k)\), denoted by \(\text{PBD}(v,b,k;\lambda,0)\), if every \(t\)-subset of \(X\) either appears together in exactly \(\lambda\) blocks of \(B\) or does not appear in any block. If \(\text{PBD}(v,b,k;x,0)\) is also a partially balanced \(s\)-design \(\text{PBD}(v,b,k;\lambda s,0)\) for \(0< s< t\), then it is called a strong partially balanced \(t\)-design and is denoted by \(\text{SPBD}(v,b,k;\lambda,0)\). SPBD is called optimal if \(b\) is the maximum number of blocks in all \(\text{SPBD}(v,b,k;\lambda,0)\). Here \(\text{OSPBD}(v,k,\lambda)\) denotes an optimal strong partially balanced 2-design. This paper deals with the case when \(v\equiv 2\pmod 4\) and the following results are proved. Result 1. There exists an \(\text{OSPBD}(v,5,1)\) for all \(v\equiv 2\pmod 4\) with \(v\geq 6\), except possibly \(v\in E\cup(22)\), where \(E=\{30,70,142,150,190,222,230, 390, 430\}\). Result 2. There exists an \(\text{OSPBD}(v,5,1)\) for all \(v\geq 5\) except when \(v\in E\cup\{22,135\}\).
scientific article; zbMATH DE number 2127151
scientific article