By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in \cite{AlfaranoBorelloNeriRavagnani2022JCTA}, we will exhibit a new class of codes with parameters $[m+2,3,m-2]_{q^m/q}$ for infinite values of $q$ and $m \geq 5$ odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.

翻译：通过利用$\mathbb{F}_{q^m}^3$中散射$\mathbb{F}_q$-子空间与$\mathbb{F}_{q^m}^{n}$（$n \geq m+2$）上最小非退化三维秩度量码之间的联系（详见\cite{AlfaranoBorelloNeriRavagnani2022JCTA}），本文为无穷多个$q$值及奇数$m \geq 5$构建了一类参数为$[m+2,3,m-2]_{q^m/q}$的新码。此外，通过研究这些散射子空间的几何结构，我们确定了相关码的秩权重分布。