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

Daniele Bartoli; Francesco Ghiandoni

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arXiv · 2403.01887

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

Daniele Bartoli·Francesco Ghiandoni

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

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Daniele Bartoli, Francesco Ghiandoni. 2024-03-04. On $3$-dimensional MRD codes of type $\langle x^{q^t},x+\delta x^{q^{2t}},G(x) \rangle$. https://arxiv.org/abs/2403.01887

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