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.

翻译：在文献[A. Neri, P. Santonastaso, F. Zullo. Extending two families of maximum rank distance codes]中，作者推广了近期在[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]中发现的二维$\mathbb{F}_{q^{2t}}$-线性MRD码族。同时，对于$t \geq 5$的情况，他们确定了该新族中元素的等价类，并给出了其中不等价码的确切数目。本文去除了$t \geq 5$的限制，完成了对该等价性问题的研究。此外，我们证明，当$t=4$时，由相关码族产生的射影直线$\mathrm{PG}(1,q^{8})$上的线性集与目前已知的任何线性集均不等价。