Constructions of Simple Cyclic $2$-designs

Abstract

In this paper, constructions of simple cyclic $2$-designs are given. As a consequence, we determined the existence of simple $2$-$(q,k,\lambda )$ designs for every admissible parameter set $(q,k,\lambda )$ where $q\le 29$ is an odd prime power, with two undecided parameter sets $(q,k,\lambda )=(29,8,6)$ and $(29,8,10)$.