Construction and nonexistence of strong external difference families

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.