  In this paper, we provide a mathematical framework for characterizing AMD
codes that are R-optimal. We introduce a new combinatorial object, the
reciprocally-weighted external difference family (RWEDF), which corresponds
precisely to an R-optimal weak AMD code. This definition subsumes known
examples of existing optimal codes, and also encompasses combinatorial objects
not covered by previous definitions in the literature. By developing structural
group-theoretic characterizations, we exhibit infinite families of new RWEDFs,
and new construction methods for known objects such as near-complete EDFs.
Examples of RWEDFs in non-abelian groups are also discussed.
