Cyclotomic Construction of Strong External Difference Families in Finite Fields

TL;DR

This paper introduces new methods for constructing strong external difference families (SEDF) and their generalizations in finite fields using cyclotomic classes, enhancing combinatorial design theory with applications in communication security.

Contribution

It presents novel constructions of SEDF, GSEDF, and BGSEDF in finite fields via cyclotomic classes, expanding the toolkit for combinatorial design in finite abelian groups.

Findings

Constructed series of SEDF, GSEDF, BGSEDF in finite fields.

Provided general methods using difference sets and partial difference sets.

Enhanced applications in communication theory and algebraic manipulation detection.

Abstract

Strong external difference family (SEDF) and its generalizations GSEDF, BGSEDF in a finite abelian group $G$ are combinatorial designs raised by Paterson and Stinson [7] in 2016 and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in $G$. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes.