Constructions of Simple Cyclic \(2\)-designs

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 \(q \leq 29\) is an odd prime power, with two undecided parameter sets \((q,k,\lambda) = (29,8,6)\) and \((29,8,10)\).