On the equivalence issue of a class of $2$-dimensional linear Maximum Rank Distance codes

TL;DR

This paper investigates the equivalence of a class of 2-dimensional linear MRD codes over finite fields, extending previous classifications to include cases where t<5 and showing new inequivalence results for t=4.

Contribution

It completes the classification of these MRD codes by removing previous restrictions and establishes that for t=4, the codes are not equivalent to previously known codes.

Findings

Extended the classification to all t values, including t<5.

Proved that for t=4, the codes are inequivalent to known codes.

Identified new inequivalence classes in the code family.

Abstract

In [A. Neri, P. Santonastaso, F. Zullo. Extending two families of maximum rank distance codes], the authors extended the family of $2$-dimensional $\mathbb{F}_{q^{2t}}$-linear MRD codes recently found in [G. Longobardi, G. Marino, R. Trombetti, Y. Zhou. A large family of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ and their associated MRD codes]. Also, for $t \geq 5$ they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction $t \geq 5$. Moreover, we prove that in the case when $t=4$, the linear sets of the projective line $\mathrm{PG}(1,q^{8})$ ensuing from codes in the relevant family, are not equivalent to any one known so far.