Weighted external difference families and R-optimal AMD codes

TL;DR

This paper introduces a new combinatorial structure called RWEDF that characterizes R-optimal AMD codes, providing a unified framework that includes known and new optimal codes, with constructions in both abelian and non-abelian groups.

Contribution

It defines RWEDF, links it to R-optimal AMD codes, and develops structural group-theoretic methods to construct infinite families of these objects.

Findings

Established a mathematical framework for R-optimal AMD codes.

Developed structural characterizations and new constructions of RWEDFs.

Extended the concept to non-abelian groups, broadening applicability.

Abstract

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.