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

The article delves into the intricate world of disjoint difference families (DDFs) within groups, beginning with the definition and necessary conditions for their existence. It then presents specific constructions for DDFs in various group orders, including the use of translations and advanced algebraic methods such as the Combinatorial Nullstellensatz. The main result is a comprehensive theorem determining the existence of DDFs under certain conditions. Additionally, the article explores applications of these constructions and provides a recursive approach to building DDFs, making it a valuable resource for researchers in combinatorics and group theory.