Strong External Difference Families and Classification of $\alpha$-valuations

TL;DR

This paper classifies all $ ext{SEDF}$s of a specific form in cyclic groups using $ ext{alpha}$-valuations of bipartite graphs, providing enumeration for small cases and exploring their equivalence and extensions to dihedral groups.

Contribution

It introduces a recursive classification of $ ext{alpha}$-valuations for constructing $ ext{SEDF}$s and enumerates all such families for small parameters, establishing their equivalence to $ ext{alpha}$-valuations.

Findings

All classified $ ext{SEDF}$s are equivalent to $ ext{alpha}$-valuations for $a \,\leq\,14$

Enumeration of all $ ext{SEDF}$s in $ ext{Z}_{a^2+1}$ for $a \,\leq\,14$

Two known dihedral group constructions are shown to be equivalent.

Abstract

One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $\alpha$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a classification theorem which shows that all such $\alpha$-valuations can be constructed recursively via a sequence of ``blow-up'' operations. We also enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\mathbb{Z}_{a^2+1}$ for $a \leq 14$ and we show that all these SEDFs are equivalent to $\alpha$-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.