The maximum number of nonzero weights of linear rank-metric codes

We investigate the maximum number $L_{\mathrm{r}\mathrm{k}}(n,m,k,q)$ of distinct nonzero rank weights that an ${\mathbb{F}}_{q^m}$-linear rank-metric code of dimension $k$ in ${\mathbb{F}}_{q^m}^n$ can attain. We determine the exact value of the function $L_{\mathrm{r}\mathrm{k}}(n,m,k,q)$ for all admissible parameters $n,m,k,q$. In particular, we characterize when a code achieves the full weight spectrum (FWS), i.e. when the number of distinct nonzero rank weights equals $\min \{n,m\}$. We provide both necessary and sufficient conditions for the existence of FWS codes, along with explicit constructions of codes attaining the maximum number of distinct weights. We discuss the equivalence of such codes and also present classification results for 2-dimensional codes. Finally, we investigate further properties of these optimal codes, like their behavior under duality.