Determine the four unresolved covering radii for extension degree six

Determine the covering radii of all one-dimensional rank-metric codes over [?] no wait

Background

For extension degree m=6, the paper identifies four parameter families whose covering radii are not determined by the preceding theoretical results: [5,1,3] codes, [6,1,3] codes, [6,1,4] codes, and [7,1,4] codes over the extension field of degree six. The authors establish only bounds for these families: the covering radius can be 3 or 4 for [5,1,3] codes, lies between 4 and 5 for [6,1,3] and [6,1,4] codes, and lies between 4 and 5 for [7,1,4] codes.

Computations for q in {2,3} give additional information, but the general problem for arbitrary prime powers q remains unresolved. In particular, for [6,1,3] codes both covering radii 4 and 5 occur for q in {2,3}, so the covering radius is not determined solely by length and minimum distance in that family.

References

For m=6 we have four cases that are still open using our results, while by Proposition~\ref{prop:boundGY} we can at least say that the covering radius of $[5,1,3]{q6/q}$ codes can be either $3$ or $4$, while for $[6,1,3]{q6/q}$ and $[6,1,4]{q6/q}$ codes we can conclude that $4\le\rho(\mathcal{C})\le 5$. As shown in the proof of Proposition~\ref{prop:tagli}, $\rho(\mathcal{C})\ge\rho(\Pi(\mathcal{C},I))$, for every $I\subseteq[n]$. Hence, if $\mathcal{C}$ is a $[7,1,4]{q6/q}$ code and we puncture it in a zero coordinate, we get $\rho(\mathcal{C})\ge 4$, hence also in this case $4\le\rho(\mathcal{C})\le 5$.

— Covering radius of rank-metric codes via covering lifts and clubs  (2610.11926 - Marino et al., 8 Oct 2026) in Section 4, subsection "Special cases" (paragraph preceding Proposition 4.3 and Table m=6)