In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence.

翻译：在秩度量码领域，最大秩距离（MRD）码是达到Singleton型界限的最优代数结构。该领域的一个主要未解问题是：判断一个MRD码能否在保持最优性的前提下被扩充为更长的码。本文研究不可扩充但未达到最大可能长度的$\mathbb{F}_{q^m}$-线性MRD码。从几何角度看，这些码对应关于超平面的在包含关系下极大但非最大维度的分散子空间。利用这一几何联系，我们引入了首个无限族的不可扩充 $[4,2,3]_{q^5/q}$ MRD码。此外，我们证明这些码在等价意义下是自对偶的。