In this work we present results on the classification of $\mathbb{F}_{q^n}$-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD codes $\mathcal{C}=\langle x^{q^t}, F(x), G(x) \rangle \subseteq \mathcal{L}_{n,q}$ of exceptional type, i.e. such that $\mathcal{C}$ is MRD over infinite many extensions of the field $\mathbb{F}_{q^n}$. These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.

翻译：本文研究了三维$\mathbb{F}_{q^n}$-线性MRD码的分类问题。特别地，利用与有限域上某些代数簇的联系，我们给出了例外型MRD码$\mathcal{C}=\langle x^{q^t}, F(x), G(x) \rangle \subseteq \mathcal{L}_{n,q}$的不存在性结果，其中该码在域$\mathbb{F}_{q^n}$的无穷多个扩域上均为MRD。这些结果部分解决了Bartoli、Zini和Zullo在2023年提出的一个猜想。