Construction and nonexistence of strong external difference families

Abstract

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters $(v, m, k, \lambda)$ of a nontrivial SEDF that is near-complete (satisfying $v=km+1$). We construct the first known nontrivial example of a $(v, m, k, \lambda)$ SEDF having $m > 2$. The parameters of this example are $(243,11,22,20)$, giving a near-complete SEDF, and its group is $\mathbb{Z}_3^5$. We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases $m=2$ and $m>2$ are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs.