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 $s$.

翻译：我们研究了对称有限幅度错误信道中$\mathbb{Z}^n$上的完美纠错码，该信道中整数向量的至多$e$个坐标可能被幅度不超过$s$的值改变。从几何角度看，此类码对应于用对称有限幅度错误球$\mathcal{B}(n,e,s,s)$对$\mathbb{Z}^n$进行铺砌。给定$n$和$s$，我们将汉明度量下Elias界的几何思想适配到该信道对应的距离度量$d_s$，并在不假设任何格结构的情况下，推导出完美码/铺砌存在性关于$e$的新必要条件。我们的主要结果根据错误幅度识别出两种不同机制。对于小错误幅度（$s \in \{1, 2\}$），我们证明若可纠正错误数量不超过$n$的特定比例，则其渐近界为$e = \mathcal{O}(\sqrt{n \log n})$。相反，对于较大幅度（$s \geq 3$），我们建立了显著更尖锐的界$e < \sqrt{12.36n}$，该界对$e$低于$n$的给定比例没有任何限制。最后，通过将方法推广至非完美码，我们推导出堆积密度的上界，表明对于可纠正线性或$Ω(\sqrt{n})$数量级错误的码，其密度受限于与错误幅度$s$成反比的因子。