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

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.