The $(n,m,k,\lambda)$-Strong External Difference Family with $m \geq 5$ Exists

TL;DR

This paper constructs a new example of a strong external difference family (SEDF) with parameters (243, 11, 22, 20) in a finite field, answering an open question about the existence of such families for m ≥ 5.

Contribution

It provides the first known example of an (n,m,k,λ)-SEDF with m ≥ 5, expanding the understanding of SEDF existence in finite groups.

Findings

Constructed an (243, 11, 22, 20)-SEDF in finite field _q.

Answered the open problem on the existence of SEDFs with m b1 5.

Extended the class of known SEDFs beyond the m=2 case.

Abstract

The notion of strong external difference family (SEDF) in a finite abelian group $(G,+)$ is raised by M. B. Paterson and D. R. Stinson [5] in 2016 and motivated by its application in communication theory to construct $R$-optimal regular algebraic manipulation detection code. A series of $(n,m,k,\lambda)$-SEDF's have been constructed in [5, 4, 2, 1] with $m=2$. In this note we present an example of (243, 11, 22, 20)-SEDF in finite field $\mathbb{F}_q$ $(q=3^5=243).$ This is an answer for the following problem raised in [5] and continuously asked in [4, 2, 1]: if there exists an $(n,m,k,\lambda)$-SEDF for $m\geq 5$.