标题： 在和度量中完美码的不存在性 显示英文标题

标题： On the non-existence of perfect codes in the sum-rank metric

摘要： 我们研究和范数度量中的完美码，这是一种同时推广了汉明度量和秩度量的度量，在多跳网络编码和空时编码中具有相关性。完美码在球体包装界达到等式，对应于环境空间中不相交度量球的划分。虽然汉明度量和秩度量中的完美码已被完全分类，但和范数度量中非平凡完美码的存在性仍 largely 开放。在本文中，我们研究和范数度量中的线性完美码。我们分析球体的几何结构并推导出它们体积的界限，说明球体包装界如何适用。对于两块空间，我们确定完美码存在的显式参数约束。对于多块空间，我们建立了在各种最小距离、可除性条件和码维度范围内的不存在结果。我们进一步提供了基于度量球体积施加的同余条件的计算证据。 显示英文摘要

摘要： We study perfect codes in the sum-rank metric, a generalization of both the Hamming and rank metrics relevant in multishot network coding and space-time coding. A perfect code attains equality in the sphere-packing bound, corresponding to a partition of the ambient space into disjoint metric balls. While perfect codes in the Hamming and rank metrics are completely classified, the existence of nontrivial perfect codes in the sum-rank metric remains largely open. In this paper, we investigate linear perfect codes in the sum-rank metric. We analyze the geometry of balls and derive bounds on their volumes, showing how the sphere-packing bound applies. For two-block spaces, we determine explicit parameter constraints for the existence of perfect codes. For multiple-block spaces, we establish non-existence results for various ranges of minimum distance, divisibility conditions, and code dimensions. We further provide computational evidence based on congruence conditions imposed by the volume of metric balls.