Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric intersecting codes exhibit substantially different behavior. It was previously shown that a nondegenerate $[n,k,d]_{q^m/q}$ rank-metric intersecting code must satisfy $2k-1 \le n \le 2m-3$, and the tightness of the upper bound was left open. Using the geometric interpretation of rank-metric codes via $q$-systems, we prove that the dual subspace associated with a rank-metric intersecting code must satisfy strong evasiveness properties. This connection allows us to derive new restrictions on the parameters of such codes and to show that the bound $n=2m-3$ can be attained only when $k=3$ and $m\ge 6$. More generally, we show that $n \leq 2m-\lfloor(k+4)/2\rfloor$. Moreover, we obtain a geometric characterization of these extremal codes in terms of scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$. As a consequence, the existence problem for $[2m-3,3,d]_{q^m/q}$ rank-metric intersecting codes is reduced to the existence of scattered subspaces of dimension $m+3$. Using known constructions of maximum scattered subspaces, we derive existence results when $m$ is even. Finally, we prove that $[6,3,3]_{q^5/q}$ rank-metric intersecting codes do not exist for any prime power $q$, thus resolving an open problem posed by Bartoli et al. in 2025.

翻译：相交码是编码理论中的一个经典对象，其秩度量类比最近被引入。尽管定义形式上与汉明度量情形平行，但秩度量相交码的结构和参数约束表现出实质不同的行为。先前已证明，非退化的 $[n,k,d]_{q^m/q}$ 秩度量相交码必须满足 $2k-1 \le n \le 2m-3$，且上界的紧性未决。利用通过 $q$-系统的秩度量码的几何解释，我们证明与秩度量相交码关联的对偶子空间必须满足强回避性质。这一联系使我们能够推导出此类码参数的新限制，并证明当且仅当 $k=3$ 且 $m\ge 6$ 时才能达到 $n=2m-3$ 的界。更一般地，我们证明 $n \leq 2m-\lfloor(k+4)/2\rfloor$。此外，我们利用 $\mathbb{F}_{q^m}^3$ 中的散射 $\mathbb{F}_q$-子空间给出了这些极值码的几何刻画。因此，$[2m-3,3,d]_{q^m/q}$ 秩度量相交码的存在性问题归结为维数为 $m+3$ 的散射子空间的存在性。利用最大散射子空间的已知构造，我们推导出当 $m$ 为偶数时的存在性结果。最后，我们证明对任意素幂 $q$，$[6,3,3]_{q^5/q}$ 秩度量相交码不存在，从而解决了 Bartoli 等人在 2025 年提出的一个开放问题。