On $3$-dimensional MRD codes of type $⟨x^{q^t},x+\delta x^{q^{2t}},G(x)⟩$

Abstract

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 $\mathcal{C}=⟨x^{q^t},F(x),G(x)⟩\subseteq {\mathcal{L}}_{n,q}$ 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.