  Strong external difference families (SEDFs) are much-studied combinatorial
objects motivated by an information security application. A well-known
conjecture states that only one abelian SEDF with more than 2 sets exists. We
show that if the disjointness condition is replaced by non-disjointness, then
abelian SEDFs can be constructed with more than 2 sets (indeed any number of
sets). We demonstrate that the non-disjoint analogue has striking differences
to, and connections with, the classical SEDF and arises naturally via another
coding application.
