Construction and nonexistence of strong external difference families

TL;DR

This paper advances the understanding of strong external difference families by characterizing their parameters, constructing a novel example with m>2, and establishing significant nonexistence results using algebraic methods.

Contribution

It provides the first known nontrivial SEDF with m>2, characterizes near-complete SEDFs, and develops a comprehensive algebraic framework distinguishing cases m=2 and m>2.

Findings

Constructed a (243,11,22,20) SEDF in  group.

Characterized parameters of near-complete SEDFs with v=km+1.

Proved nonexistence results narrowing possible SEDF parameters.

Abstract

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters $(v, m, k, \lambda)$ of a nontrivial SEDF that is near-complete (satisfying $v=km+1$). We construct the first known nontrivial example of a $(v, m, k, \lambda)$ SEDF having $m > 2$. The parameters of this example are $(243,11,22,20)$, giving a near-complete SEDF, and its group is $\mathbb{Z}_3^5$. We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases $m=2$ and $m>2$ are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs.