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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.13477 |
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| _version_ | 1866911386937851904 |
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| author | Guan, Zhihao Wei, Hengjia |
| author_facet | Guan, Zhihao Wei, Hengjia |
| contents | We study perfect error-correcting codes in $\mathbb{Z}^n$ for the symmetric limited-magnitude error channel, where at most $e$ coordinates of an integer vector may be altered by a value whose magnitude is at most $s$. Geometrically, such codes correspond to tilings of $\mathbb{Z}^n$ by the symmetric limited-magnitude error ball $\mathcal{B}(n,e,s,s)$. Given $n$ and $s$, we adapt the geometric ideas underlying the Elias bound for the Hamming metric to the distance $d_s$ tailed for this channel, and derive new necessary conditions on $e$ for the existence of perfect codes / tilings, without assuming any lattice structure. Our main results identify two distinct regimes depending on the error magnitude. For small error magnitudes ($s \in \{1, 2\}$), we prove that if the number of correctable errors does not exceed a certain fraction of $n$, then it is asymptotically bounded by $e = \mathcal{O}(\sqrt{n \log n})$. In contrast, for larger magnitudes ($s \geq 3$), we establish a significantly sharper bound of $e < \sqrt{12.36n}$, which holds without any restriction on $e$ being below a given fraction of $n$. Finally, by extending our method to non-perfect codes, we derive an upper bound on packing density, showing that for codes correcting a linear or $Ω(\sqrt{n})$ number of errors, the density is bounded by a factor inversely proportional to the error magnitude $s$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13477 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Elias-type Bounds for Codes in the Symmetric Limited-Magnitude Error Channel Guan, Zhihao Wei, Hengjia Information Theory We study perfect error-correcting codes in $\mathbb{Z}^n$ for the symmetric limited-magnitude error channel, where at most $e$ coordinates of an integer vector may be altered by a value whose magnitude is at most $s$. Geometrically, such codes correspond to tilings of $\mathbb{Z}^n$ by the symmetric limited-magnitude error ball $\mathcal{B}(n,e,s,s)$. Given $n$ and $s$, we adapt the geometric ideas underlying the Elias bound for the Hamming metric to the distance $d_s$ tailed for this channel, and derive new necessary conditions on $e$ for the existence of perfect codes / tilings, without assuming any lattice structure. Our main results identify two distinct regimes depending on the error magnitude. For small error magnitudes ($s \in \{1, 2\}$), we prove that if the number of correctable errors does not exceed a certain fraction of $n$, then it is asymptotically bounded by $e = \mathcal{O}(\sqrt{n \log n})$. In contrast, for larger magnitudes ($s \geq 3$), we establish a significantly sharper bound of $e < \sqrt{12.36n}$, which holds without any restriction on $e$ being below a given fraction of $n$. Finally, by extending our method to non-perfect codes, we derive an upper bound on packing density, showing that for codes correcting a linear or $Ω(\sqrt{n})$ number of errors, the density is bounded by a factor inversely proportional to the error magnitude $s$. |
| title | Elias-type Bounds for Codes in the Symmetric Limited-Magnitude Error Channel |
| topic | Information Theory |
| url | https://arxiv.org/abs/2601.13477 |