Code-based $[3,1]$-avoiders in finite affine spaces $\mathrm{AG}(n,2)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916767183405056 |
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| author | Kovács, Benedek |
| author_facet | Kovács, Benedek |
| contents | The author, together with Nagy, studied the following problem on unavoidable intersections of given size in binary affine spaces. Given an $m$-element set $S\subseteq \mathbb{F}_2^n$, is there guaranteed to be a $[k,t]$-flat, that is, a $k$-dimensional affine subspace of $\mathbb{F}_2^n$ containing exactly $t$ points of $S$? Such problems can be viewed as generalizations of the cap set problem over the binary field. They conjectured that for every fixed pair $(k,t)$ with $k\ge 1$ and $0\le t\le 2^k$, the density of values $m\in \{0,...,2^n\}$ for which a $[k,t]$-flat is guaranteed tends to $1$.
In this paper, motivated by the study of the smallest open case $(k,t)=(3,1)$, we present explicit constructions of sets in $\mathbb{F}_2^n$ avoiding $[k,1]$-flats for exponentially many sizes. These sets rely on carefully constructed binary linear codes, whose weight enumerators determine the size of the construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Code-based $[3,1]$-avoiders in finite affine spaces $\mathrm{AG}(n,2)$ Kovács, Benedek Combinatorics 51E21, 51E22, 20G15, 94B05 The author, together with Nagy, studied the following problem on unavoidable intersections of given size in binary affine spaces. Given an $m$-element set $S\subseteq \mathbb{F}_2^n$, is there guaranteed to be a $[k,t]$-flat, that is, a $k$-dimensional affine subspace of $\mathbb{F}_2^n$ containing exactly $t$ points of $S$? Such problems can be viewed as generalizations of the cap set problem over the binary field. They conjectured that for every fixed pair $(k,t)$ with $k\ge 1$ and $0\le t\le 2^k$, the density of values $m\in \{0,...,2^n\}$ for which a $[k,t]$-flat is guaranteed tends to $1$. In this paper, motivated by the study of the smallest open case $(k,t)=(3,1)$, we present explicit constructions of sets in $\mathbb{F}_2^n$ avoiding $[k,1]$-flats for exponentially many sizes. These sets rely on carefully constructed binary linear codes, whose weight enumerators determine the size of the construction. |
| title | Code-based $[3,1]$-avoiders in finite affine spaces $\mathrm{AG}(n,2)$ |
| topic | Combinatorics 51E21, 51E22, 20G15, 94B05 |
| url | https://arxiv.org/abs/2505.24072 |