The existence of cyclic (v,4,1)-designs

Abstract

Even though Peltesohn proved that a cyclic (v,3,1)-design exists if and only if $v\equiv 1,3(\mathrm{mod}6)$ as early as 1939, the problem of determining the spectrum of cyclic (v,k,1)-designs with k>3 is far from being settled, even for k=4. This paper shows that a cyclic (v,4,1)-design exists if and only if $v\equiv 1,4(\mathrm{mod}12)$ and $v\notin \{16,25,28\}$.