Super-Simple Balanced Incomplete Block Designs With Block Size 5 and Index 8

Abstract

A design is said to be super-simple if the intersection of any two blocks contains at most two elements. In the statistical design of experiments, super-simple designs yield samples in which the maximum pairwise block intersection is minimized. Moreover, such designs play a valuable role in the construction of codes, including optical orthogonal codes and superimposed codes, other combinatorial designs such as group divisible designs and pairwise balanced designs, and can be used to determine some exact Zarankiewicz numbers. In this paper, we investigate the existence of a super-simple $\left(v,5,8\right)$-BIBD and show that such a design exists if and only if $v\equiv 0,1\left(\mathrm{mod}5\right)$ and $v\ge 26$.