On 3-dimensional MRD codes of type \(\langle X^{q^t},X+\delta X^{q^{2t}},G(X) \rangle \)

In this work we present results on the classification of \(\mathbb {F}_{q^n}\) ${\mathbb{F}}_{q^n}$ -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 \(\mathcal {C}=\langle X^{q^t}, F(X), G(X) \rangle \subseteq \mathcal {L}_{n,q}\) $\mathcal{C}=⟨X^{q^t},F(X),G(X)⟩\subseteq {\mathcal{L}}_{n,q}$ of exceptional type, i.e. such that \(\mathcal {C}\) $\mathcal{C}$ is MRD over infinitely many extensions of the base field. These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.