Resolvable balanced incomplete block designs with block size 8
A balanced incomplete block design (BIBD) \(D\) is a collection \(B\) of subsets (blocks) taken from a finite set \(V\) of \(v\) elements with the properties: (1) every pair of distinct elements of \(V\) is contained in precisely \(\lambda\) blocks of \(B\), and (2) every block contains exactly \(k\) elements. We denote such a design as a \((v,k,\lambda)\)-BIBD. A \((v,k,\lambda)\)-BIBD is said to be resolvable if the blocks can be partitioned in classes \(R_1,R_2,\dots,R_r\) (resolution classes) where \(r=\lambda (v-1)/(k-1)\) such that each element of \(D\) is contained in precisely one block of each class. The resolution classes \(R_1,R_2,\dots,R_r\) form a resolution of \(D\). A necessary condition for the existence of a resolvable \((v,8,1)\)-BIBD is that \(v\equiv 8\pmod{56}\). With the possible exception of 66 values of \(v\), this condition is shown to be sufficient. The largest exceptional value of \(v\) is 24480.
scientific article; zbMATH DE number 1028922