2025/10/23 by Liu, Hengrui, Feng, Tao, Wang, Xiaomiao +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.20446
Let v be a positive odd integer. A (v,k,λ)-perfect difference family (PDF) is a collection F of k-subsets of \0,1,…,v-1\ such that the multiset \bigcupF∈ F\x-y : x,y∈ F, x>y\ covers each element of \1,2,…,(v-1)/2\ exactly λ 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 Erdős. In this paper, we prove that a (v,4,λ)-PDF exists if and only if λ(v-1) ≡ 0 \pmod12, v ≥ 13, and (v,λ) ∉ \(25,1),(37,1)\. 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 orthogonal codes for DNA origami, difference triangle sets, additive sequences of permutations, and graceful graph labelings. To establish our main result, we introduce a new concept termed a layered difference family. This concept provides a powerful and unified perspective that not only facilitates our proof of the main theorem but also simplifies recent existence proofs for various cyclic difference packings.