MRD-codes arising from the trinomial $x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$

TL;DR

This paper extends the construction of certain MRD-codes from trinomial polynomials over finite fields, demonstrating their existence in all odd cases and establishing their novelty compared to known codes.

Contribution

It shows that MRD-codes derived from a specific trinomial form exist in all odd cases, not just those previously proven, and are inequivalent to known codes.

Findings

MRD-codes exist for all odd q cases from the trinomial form

These codes are not equivalent to previously known MRD-codes

Corresponding linear sets are not PΓL-equivalent to known sets

Abstract

In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of $\mathbb{F}_{q^6}[x]$, when $q$ is odd and $q\equiv 0,\pm 1\pmod 5$, has been proved. In this paper we show that this family produces $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with the same properties, also in the remaining $q$ odd cases, but not in the $q$ even case. These MRD-codes are not equivalent to the previously known MRD-codes. We also prove that the corresponding maximum scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not $\mathrm{P}\Gamma\mathrm{L}(2,q^6)$-equivalent to any previously known scattered linear set.