The covering radius of a code is the least positive integer \(\rho\) such that the union of the spheres of radius \(\rho\) about each codeword is equal to the full ambient space. It gives a measure of its error-correcting capabilities via a determination of the maximal weight of a correctable error.
The rank distance between \(u, v\in\mathbb{F}_{q^m}^n\) is defined to be the \(\mathbb{F}_q\)-dimension of the vector space, i.e. \(d_{\text{rk}}(u, v)=\text{dim}\mathbb{F}_q \left\langle u_i-v_i \vert 1\leq i \leq n\right\rangle_{\mathbb{F}_q}.\) An \([n, k, d]_{q^m/q}\) rank-metric code \(C\) is a \(k\)-dimensional \(\mathbb{F}_{q^m}\)-subspace of \(\mathbb{F}_{q^m}^n\) such that \(d = d_{\text{rk}}(C)= \text{min}\{d_{\text{rk}}(c, c') \vert c, c'\in C, c\neq c'\}.\) The authors are studying the covering radius problems in the rank metric settings from a geometric viewpoint. To achieve their goals, the notion of a \([n, k]_{q^m/q}\) rank-\(\rho\)-saturating system in correspondence with a rank-metric covering code is introduced.
Let \(s_{q^m/q}(k, \rho)\) denotes the minimum \(\mathbb{F}_q\)-dimension of a rank-\(\rho\)-saturating system in \(\mathbb{F}_{q^m}^k.\) It’s shown that this number for \(q>2\) satisfies the following bounds: \(\left\lceil\cfrac{mk}{\rho}\right\rceil-m+\rho\leq s_{q^m/q}(k, \rho)\leq m(k-\rho)+\rho,\) and the lower bound is sharpened when \(q=2.\)
Two different approaches are shown: one construction steaming from linear cutting blocking sets, and the other uses subgeometries. Finally, eight cases are shown, for which \(s_{q^m/q}(k, \rho)\) is completely determined, including when \(\rho=1\) or \(k\); when \(\rho=2, k=3\) and \(m=2r\) or 10; and when \(m=k=2r\) and \(\rho=k-1.\)