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

TL;DR

This paper characterizes the existence of cyclic (v,4,1)-designs, establishing precise modular conditions and excluding specific values, thus advancing the understanding of combinatorial design existence.

Contribution

It provides a complete characterization of when cyclic (v,4,1)-designs exist, filling a gap in the theory beyond the well-understood (v,3,1)-designs.

Findings

Cyclic (v,4,1)-designs exist iff v ≡ 1,4 mod 12

Designs do not exist for v=16,25,28

Existence characterized by modular conditions

Abstract

Even though Peltesohn proved that a cyclic (v,3,1)-design exists if and only if $v\equiv 1,3\pmod{6}$ 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 $v\equiv 1,4\pmod{12}$ and $v\not\in\{16,25,28\}$.