Existence and Non-Existence Results for Strong External Difference Families

TL;DR

This paper investigates the existence conditions of strong external difference families (SEDFs), providing new necessary criteria, especially for the case where lambda equals 2, and linking certain SEDFs to Paley partial difference sets.

Contribution

It introduces new necessary conditions for the existence of (n; m; k; lambda)-SEDFs and characterizes partition type SEDFs for m=2, connecting them to Paley partial difference sets.

Findings

Established new necessary conditions for SEDFs existence.

Provided a structural characterization for partition type SEDFs when m=2.

Extended results to generalized SEDFs with non-trivial conditions.

Abstract

We consider strong external difference families (SEDFs); these are external difference families satisfying additional conditions on the patterns of external diferences that occur, and were first defined in the context of classifying optimal strong algebraic manipulation detection codes. We establish new necessary conditions for the existence of (n; m; k; lambda)-SEDFs; in particular giving a near-complete treatment of the lambda = 2 case. For the case m = 2, we obtain a structural characterization for partition type SEDFs (of maximum possible k and lambda), showing that these correspond to Paley partial difference sets. We also prove a version of our main result for generalized SEDFs, establishing non-trivial necessary conditions for their existence.