Covering radius of rank-metric codes via covering lifts and clubs

@misc{marino2026covering,
  title = {{Covering radius of rank-metric codes via covering lifts and clubs}},
  author = {Giuseppe Marino and Alessandro Neri and Alessia Vallefuoco},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11926},
  eprint = {2610.11926},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11926},
}

TY  - UNPB
TI  - Covering radius of rank-metric codes via covering lifts and clubs
AU  - Giuseppe Marino
AU  - Alessandro Neri
AU  - Alessia Vallefuoco
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11926
UR  - https://arxiv.org/abs/2610.11926
AB  - We introduce a generator-matrix-based geometric approach to the covering radius of 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(𝒞)≤ 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(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∈{3,4,5}. For m=6 and every q we provide constructions and, for q∈{2,3}, computational results showing that $1$-dimensional rank-metric codes with the same length and minimum distance can have different covering radii.
ER  -