Innovation
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rank-metric codes
covering radius
covering lifts
clubs
q-system
Covering radius of rank-metric codes via covering lifts and clubs
🤖 AI Summary
This study addresses the exact computation and bounding of the covering radius for rank-metric codes. Methodologically, by leveraging generator matrix geometry and q-system theory, it introduces the novel concept of "covering lifting," which recovers the covering radius via hyperplane weights and reduces the one-dimensional code problem to characterizing linear maps and evasive subspaces. The main contributions include establishing an exact criterion and a general upper bound for the maximum covering radius, as well as revealing intrinsic connections between one-dimensional codes and club structures. Furthermore, this work determines the covering radii of one-dimensional codes for extension degrees three through five, uncovering a new phenomenon wherein codes sharing identical parameters can exhibit distinct covering radii under specific configurations.
📝 Abstract
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.
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