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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.11783 |
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| _version_ | 1866911957202763776 |
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| author | Thornburgh, Darrion |
| author_facet | Thornburgh, Darrion |
| contents | A Sidon set $S$ in $\mathbb{F}_2^n$ is a set such that the pairwise sums of distinct points are all distinct. The exclude points of a Sidon set $S$ are the sums of three distinct points in $S$, and the exclude multiplicity of a point in $\mathbb{F}_2^n \setminus S$ is the number of such triples in $S$ it is equal to. We call the function $d_S \colon \mathbb{F}_2^n \setminus S \to \mathbb{Z}_{\geq 0}$ taking points in $\mathbb{F}_2^n \setminus S$ to their exclude multiplicity the exclude distribution of $S$. We say that $d_S$ is uniform on $\mathcal{P}$ if $\mathcal{P}$ is an equally-sized partition $\mathcal{P}$ of $\mathbb{F}_2^n \setminus S$ such that $d_S$ takes the same values an equal number of times on every element of $\mathcal{P}$. In this paper, we use APN plateaued functions with all component functions unbalanced to construct Sidon sets $S$ in $(\mathbb{F}_2^n)^2$ whose exclude distributions are uniform on natural partitions of $(\mathbb{F}_2^n)^2 \setminus S$ into $2^n$ elements. We use this result and a result of Carlet to determine exactly what values the exclude distributions of the graphs of the Gold and Kasami functions take and how often they take these values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform exclude distributions of Sidon sets Thornburgh, Darrion Combinatorics A Sidon set $S$ in $\mathbb{F}_2^n$ is a set such that the pairwise sums of distinct points are all distinct. The exclude points of a Sidon set $S$ are the sums of three distinct points in $S$, and the exclude multiplicity of a point in $\mathbb{F}_2^n \setminus S$ is the number of such triples in $S$ it is equal to. We call the function $d_S \colon \mathbb{F}_2^n \setminus S \to \mathbb{Z}_{\geq 0}$ taking points in $\mathbb{F}_2^n \setminus S$ to their exclude multiplicity the exclude distribution of $S$. We say that $d_S$ is uniform on $\mathcal{P}$ if $\mathcal{P}$ is an equally-sized partition $\mathcal{P}$ of $\mathbb{F}_2^n \setminus S$ such that $d_S$ takes the same values an equal number of times on every element of $\mathcal{P}$. In this paper, we use APN plateaued functions with all component functions unbalanced to construct Sidon sets $S$ in $(\mathbb{F}_2^n)^2$ whose exclude distributions are uniform on natural partitions of $(\mathbb{F}_2^n)^2 \setminus S$ into $2^n$ elements. We use this result and a result of Carlet to determine exactly what values the exclude distributions of the graphs of the Gold and Kasami functions take and how often they take these values. |
| title | Uniform exclude distributions of Sidon sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.11783 |