Loosely related to this: Bounding the length in a module of evaluated skew polynomials

Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \geq n$ and let $d$ be the minimum rank of a nonzero matrix in $C$, then we have the well-known Singleton bound

$$\operatorname{dim}_{\mathbb{F}_q}(C) \leq m(n-d+1),$$

which also immediately gives an upper bound for the cardinality of $C$. I am trying to generalise this inequality to matrices over general finite principal ideal rings and replace the minimum rank $d$ with the minimum length (over $R$) of the module generated by the rows of a matrix. Now we can obtain an upper bound about the cardinality of $C$ in the latter case (see e.g. Singleton Bounds for Codes over Finite Rings,Theorem 1), but I am currently stuck with trying to obtain an upper bound for the length (instead of the cardinality) of $C$ as an $R$-module, depending on $m,n,d$ and $R$.

Is there an existing result related to this? I have found precisely $0$ articles or books mentioning any results about the length of the row space of a matrix over a ring. Is the situation maybe clearer when $R$ is local?



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