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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2501.08636 |
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| _version_ | 1866912317042589696 |
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| author | Guan, Zhihao Wei, Hengjia Xiang, Ziqing |
| author_facet | Guan, Zhihao Wei, Hengjia Xiang, Ziqing |
| contents | Limited-magnitude errors modify a transmitted integer vector in at most $t$ entries, where each entry can increase by at most $\kp$ or decrease by at most $\km$. This channel model is particularly relevant to applications such as flash memories and DNA storage. A perfect code for this channel is equivalent to a tiling of $\Z^n$ by asymmetric limited-magnitude balls $\cB(n,t,\kp,\km)$. In this paper, we focus on the case where $t=2$ and $\km=\kp-1$, and we derive necessary conditions on $m$ and $n$ for the existence of a lattice tiling of $\cB(n,2,m,m-1)$. Specifically, we prove that if such a tiling exists, then either $4\leq m \leq 512$ and $n<7.23m+4$, or $m>512$ and $n<4m$. In particular, for $m=2$ and $m=3$, we show that no lattice tiling of $\cB(n,2,2,1)$ or $\cB(n,2,3,2)$ exists for any $n\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Lattice Tilings of Asymmetric Limited-Magnitude Balls $\cB(n,2,m,m-1)$ Guan, Zhihao Wei, Hengjia Xiang, Ziqing Combinatorics Limited-magnitude errors modify a transmitted integer vector in at most $t$ entries, where each entry can increase by at most $\kp$ or decrease by at most $\km$. This channel model is particularly relevant to applications such as flash memories and DNA storage. A perfect code for this channel is equivalent to a tiling of $\Z^n$ by asymmetric limited-magnitude balls $\cB(n,t,\kp,\km)$. In this paper, we focus on the case where $t=2$ and $\km=\kp-1$, and we derive necessary conditions on $m$ and $n$ for the existence of a lattice tiling of $\cB(n,2,m,m-1)$. Specifically, we prove that if such a tiling exists, then either $4\leq m \leq 512$ and $n<7.23m+4$, or $m>512$ and $n<4m$. In particular, for $m=2$ and $m=3$, we show that no lattice tiling of $\cB(n,2,2,1)$ or $\cB(n,2,3,2)$ exists for any $n\geq 3$. |
| title | On Lattice Tilings of Asymmetric Limited-Magnitude Balls $\cB(n,2,m,m-1)$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.08636 |