Digraph-defined external difference families and new circular external difference families

Abstract

External difference families (EDFs) are combinatorial objects which were introduced in the early 2000s, motivated by information security applications such as the construction of AMD codes. Various generalizations have since been defined and investigated, in particular strong external difference families (SEDFs) and circular external difference families (CEDFs). In this paper, we present a framework based on graphs and digraphs which offers a new unified way to view these structures, and leads to natural new research questions. We present constructions and structural results about these digraph-defined EDFs, and we obtain new explicit constructions for infinite families of CEDFs, in particular $(ml^2+1,m,l,1)$-CEDFs. Our techniques include cyclotomy in finite fields and direct constructions in cyclic groups and direct products of cyclic groups. We construct the first infinite family of such CEDFs in non-cyclic abelian groups; these have odd values of $m$ and $l$. We also present the first CEDF in a non-abelian group.