Perfect difference families, perfect systems of difference sets and their applications
Hengrui Liu, Tao Feng, Xiaomiao Wang, Menglong Zhang

TL;DR
This paper characterizes the existence of perfect difference families with parameters (v,4,λ), resolving a long-standing conjecture and introducing layered difference families to unify and simplify related combinatorial existence proofs.
Contribution
The paper proves a complete characterization of (v,4,λ)-perfect difference families, solving a 50-year-old conjecture, and introduces layered difference families to streamline existence proofs.
Findings
Existence of (v,4,λ)-PDF iff λ(v-1) ≡ 0 mod 12, v ≥ 13, and not (25,1) or (37,1)
Introduces layered difference families as a new concept for unified analysis
Simplifies proofs of cyclic difference packings
Abstract
Let be a positive odd integer. A -perfect difference family (PDF) is a collection of -subsets of such that the multiset covers each element of exactly times. Perfect difference families are a special class of perfect systems of difference sets. They were introduced by Bermond, Kotzig, and Turgeon in the 1970s, following a problem suggested by Erd\H{o}s. In this paper, we prove that a -PDF exists if and only if , , and . This result resolves a nearly 50-year-old conjecture posed by Bermond. Perfect difference families find applications in radio astronomy, optical orthogonal codes for optical code-division multiple access systems, geometric…
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Taxonomy
Topicsgraph theory and CDMA systems · Limits and Structures in Graph Theory · Advanced Graph Theory Research
