发表机构
Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II（那不勒斯费德里科二世大学数学与应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出基于生成矩阵的几何方法求$\text{F}_{q^m}$线性秩距离码的覆盖半径，针对1维码等情形给出判据、等价解释及构造，发现相同参数的1维秩距离码覆盖半径可不同。
AI 中文摘要
我们针对$\boldsymbol{\text{F}}_{q^m}$线性秩距离码的覆盖半径，引入一种基于生成矩阵的几何方法。从与该码关联的$q$-系统出发，我们定义了覆盖提升的概念，并证明覆盖半径可从这类提升的超平面权重中恢复。这给出了一个精确判据，根据码的长度和有效长度判断覆盖半径是否达到其最大可能值，同时在其余所有情形给出上界$\rho(\boldsymbol{\text{C}})\boldsymbol{\text{≤ min}}\boldsymbol{\text{\textit{m},n}}\boldsymbol{-1}$。随后我们将该方法专门应用于1维码，此时有效长度与最小秩距离一致。在该情形下，达到$\rho(\boldsymbol{\text{C}})\boldsymbol{= min}\boldsymbol{\text{\textit{m},n}}\boldsymbol{-1}$的码与作为覆盖提升的俱乐部集相关。更一般地，我们通过映射到合适商空间的线性映射刻画该情形，得到关于广义规避子空间和辅助矩阵秩距离码的等价解释。这些结果确定了最小距离若干边界值以及所有扩张度$\boldsymbol{\text{m}}\boldsymbol{\text{∈}}\boldsymbol{\text{\textit{3,4,5}}}$的1维码的覆盖半径。对于$\boldsymbol{\text{m}}\boldsymbol{\text{=6}}$和任意$\boldsymbol{\text{q}}$，我们给出构造；对于$\boldsymbol{\text{q}}\boldsymbol{\text{∈}}\boldsymbol{\text{\textit{2,3}}}$，计算结果表明，具有相同长度和最小距离的1维秩距离码可具有不同的覆盖半径。
英文摘要
We introduce a generator-matrix-based geometric approach to the covering radius of $\mathbb F_{q^m}$-linear rank-metric codes. Starting from the $q$-system associated with the code, we define the notion of covering lift and show that the covering radius can be recovered from the hyperplane weights of such lifts. This yields an exact criterion, in terms of the length and effective length of the code, for the covering radius to attain its largest possible value, together with the upper bound $ρ(\mathcal{C})\le \min\{m,n\}-1$ in all remaining cases. We then specialize our approach to $1$-dimensional codes, for which the effective length coincides with the minimum rank distance. In this setting, codes attaining $ρ(\mathcal{C})= \min\{m,n\}-1$ are related to covering lifts that are clubs. More generally, we characterize this case through linear maps into suitable quotient spaces, obtaining equivalent interpretations in terms of generalised evasive subspaces and auxiliary matrix rank-metric codes. These results determine the covering radius for several boundary values of the minimum distance and for all $1$-dimensional codes with extension degree $m\in\{3,4,5\}$. For $m=6$ and every $q$ we provide constructions and, for $q\in\{2,3\}$, computational results showing that $1$-dimensional rank-metric codes with the same length and minimum distance can have different covering radii.
