Strong external difference families and classification of α-valuations

One method of constructing (a2 + 1, 2, a, 1)-SEDFs (i.e., strong external difference families) in Za2+1 makes use of α-valuations of complete bipartite graphs Ka,a. We explore this approach and we provide a classification theorem which shows that all such α-valuations can be constructed recursively via a sequence of “blow-up” operations. We also enumerate all (a2 +1, 2, a, 1)-SEDFs in Za2+1 for a ≤ 14 and we show that all these SEDFs are equivalent to α-valuations via affine transformations. Whether this holds for all a > 14 as well is an interesting open problem. We also study SEDFs in dihedral groups, where we show that two known constructions are equivalent.