The existence of \((\mathbb {Z}_v,4,1)\)-disjoint difference families

This paper shows that a \((\mathbb {Z}_v,4,1)\) $({\mathbb{Z}}_v,4,1)$ -disjoint difference family exists if and only if \(v\equiv 1\pmod {12}\) $v\equiv 1(mod12)$ and \(v\ne 25\) $v\ne 25$ by giving suitable translations of base blocks of a \((\mathbb {Z}_v,4,1)\) $({\mathbb{Z}}_v,4,1)$ -cyclic difference family. The Combinatorial Nullstellensatz finds its application in constructing disjoint difference families.