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Digraph-defined external difference families and new circular external difference families

Sophie Huczynska, Christopher Jefferson, Struan McCartney

TL;DR

This work introduces a graph-theoretic, digraph-based framework for external difference families (EDFs) and their variants, unifying EDFs, SEDFs, and CEDFs under the notion of H-defined EDFs. It develops constructions for digraph-defined EDFs across complete graphs, cycles, and complete bipartite graphs, leveraging cyclotomy in finite fields and direct product structures, and obtains new infinite families of circular EDFs, including $(ml^2+1,m,l,1)$-CEDFs, as well as the first CEDF in a non-abelian group. Notably, it provides an infinite family of CEDFs in non-cyclic abelian groups with odd $m,l$, and the first CEDF in a non-abelian group, along with inequivalence results for CEDFs in cyclic groups and a complete bipartite EDF/GSEDF/GEDF correspondence. The results open multiple research directions, including broader graph families for EDFs, non-disjoint and unequal-size variants, and detailed investigations into how the underlying group structure affects feasibility and parameters.

Abstract

External difference families (EDFs) are combinatorial objects which were introduced in the early 2000s, motivated by information security applications such as the construction of AMD codes. Various generalizations have since been defined and investigated, in particular strong external difference families (SEDFs) and circular external difference families (CEDFs). In this paper, we present a framework based on graphs and digraphs which offers a new unified way to view these structures, and leads to natural new research questions. We present constructions and structural results about these digraph-defined EDFs, and we obtain new explicit constructions for infinite families of CEDFs, in particular $(ml^2+1,m,l,1)$-CEDFs. Our techniques include cyclotomy in finite fields and direct constructions in cyclic groups and direct products of cyclic groups. We construct the first infinite family of such CEDFs in non-cyclic abelian groups; these have odd values of $m$ and $l$. We also present the first CEDF in a non-abelian group.

Digraph-defined external difference families and new circular external difference families

TL;DR

This work introduces a graph-theoretic, digraph-based framework for external difference families (EDFs) and their variants, unifying EDFs, SEDFs, and CEDFs under the notion of H-defined EDFs. It develops constructions for digraph-defined EDFs across complete graphs, cycles, and complete bipartite graphs, leveraging cyclotomy in finite fields and direct product structures, and obtains new infinite families of circular EDFs, including -CEDFs, as well as the first CEDF in a non-abelian group. Notably, it provides an infinite family of CEDFs in non-cyclic abelian groups with odd , and the first CEDF in a non-abelian group, along with inequivalence results for CEDFs in cyclic groups and a complete bipartite EDF/GSEDF/GEDF correspondence. The results open multiple research directions, including broader graph families for EDFs, non-disjoint and unequal-size variants, and detailed investigations into how the underlying group structure affects feasibility and parameters.

Abstract

External difference families (EDFs) are combinatorial objects which were introduced in the early 2000s, motivated by information security applications such as the construction of AMD codes. Various generalizations have since been defined and investigated, in particular strong external difference families (SEDFs) and circular external difference families (CEDFs). In this paper, we present a framework based on graphs and digraphs which offers a new unified way to view these structures, and leads to natural new research questions. We present constructions and structural results about these digraph-defined EDFs, and we obtain new explicit constructions for infinite families of CEDFs, in particular -CEDFs. Our techniques include cyclotomy in finite fields and direct constructions in cyclic groups and direct products of cyclic groups. We construct the first infinite family of such CEDFs in non-cyclic abelian groups; these have odd values of and . We also present the first CEDF in a non-abelian group.
Paper Structure (13 sections, 15 theorems, 32 equations, 4 tables)

This paper contains 13 sections, 15 theorems, 32 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. A new viewpoint
  4. $H$-defined EDFs when $H$ is complete
  5. External difference families
  6. Tournament-defined EDFs
  7. $H$-defined EDFs when $H$ is a cycle or union of cycles
  8. Circular external difference families
  9. CEDFs in abelian non-cyclic groups and non-abelian groups
  10. Inequivalent CEDFs in cyclic groups
  11. EDFs defined by undirected cycles
  12. $H$-defined EDFs when $H$ is complete bipartite
  13. Further work

Key Result

Lemma 2.4

Let $\mathcal{A}=\{A_1, \ldots, A_m\}$ be a disjoint collection of $l$-subsets of a group $G$. Suppose $\mathcal{A}$ is a $c$-CEDF. Then, if $\gcd(c,m)=d$, the multiset in the definition may be written as a disjoint union of $d$ multiset unions, each involving $m/d$ sets, as follows (where all indic

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Example 3.1
  • Definition 3.2
  • Example 3.3
  • Remark 3.4
  • ...and 37 more