Existence and non-existence results for strong external difference families

Abstract

We consider strong external difference families (SEDFs); these are external difference families satisfying additional conditions on the patterns of external differences that occur, and were first defined in the context of classifying optimal strong algebraic manipulation detection codes. We establish new necessary conditions for the existence of (n,m,k,λ)-SEDFs; in particular giving a near-complete treatment of the λ=2 case. For the case m=2, we obtain a structural characterization for partition type SEDFs (of maximum possible k and λ), showing that these correspond to Paley partial difference sets. We also prove a version of our main result for generalized SEDFs, establishing non-trivial necessary conditions for their existence.