Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \operatorname{Mat}_{m \times n}(R)$ (1)). Let $M$ be an $R$-module and $\lambda_R(M)$ its length over $R$.

Let $S[x, \sigma]$ the ring of skew polynomials over $S$ with automorphism $\sigma$. Let $g = (g_1, \dots , g_n) \in S^n$, where $\{g_1, \dots , g_n\}$ is an $R$-lin.indep. set of elements of $S$, and $\operatorname{Gab}_k(g)=\{f(g) \ | \ f \in S[x, \sigma], \ \operatorname{deg}(f) <k\}$, so in particular, by (1), $\operatorname{Gab}_k(g)$ can be viewed as an $R$-module of $m \times n$ matrices. Furthermore, it can be shown that $\operatorname{Gab}_k(g)$ has a generator matrix $\textbf{G}$ whose $k$ rows are $\sigma^i(g_1) \ \sigma^i(g_2) \ \dots \ \sigma^i(g_n)$, for $i=0, \dots, k-1$.

Is there a way to compute $\operatorname{min}\{\lambda_R(\operatorname{rowsp}(A)) \ | \ A \in \operatorname{Gab}_k(g)\}$, where $\lambda_R(\operatorname{rowsp}(A))$ denotes the length of the module generated by the rows of the matrix $A$? This is closely related to the well-known results about Gabidulin codes over fields, but the techniques used for the latter cannot be applied in this case. I feel that it should not be very difficult, but I don't have much expertise with skew polynomials.



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