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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.20446 |
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| _version_ | 1866909865197174784 |
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| author | Liu, Hengrui Feng, Tao Wang, Xiaomiao Zhang, Menglong |
| author_facet | Liu, Hengrui Feng, Tao Wang, Xiaomiao Zhang, Menglong |
| contents | Let $v$ be a positive odd integer. A $(v,k,λ)$-perfect difference family (PDF) is a collection $\mathcal{F}$ of $k$-subsets of $\{0,1,\ldots,v-1\}$ such that the multiset $\bigcup_{F\in \mathcal{F}}\{x-y : x,y\in F, x>y\}$ covers each element of $\left\{1,2,\ldots,(v-1)/2\right\}$ 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 $λ(v-1) \equiv 0 \pmod{12}$, $v \geq 13$, and $(v,λ) \notin \{(25,1),(37,1)\}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect difference families, perfect systems of difference sets and their applications Liu, Hengrui Feng, Tao Wang, Xiaomiao Zhang, Menglong Combinatorics Let $v$ be a positive odd integer. A $(v,k,λ)$-perfect difference family (PDF) is a collection $\mathcal{F}$ of $k$-subsets of $\{0,1,\ldots,v-1\}$ such that the multiset $\bigcup_{F\in \mathcal{F}}\{x-y : x,y\in F, x>y\}$ covers each element of $\left\{1,2,\ldots,(v-1)/2\right\}$ 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 $λ(v-1) \equiv 0 \pmod{12}$, $v \geq 13$, and $(v,λ) \notin \{(25,1),(37,1)\}$. 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. |
| title | Perfect difference families, perfect systems of difference sets and their applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.20446 |