  By using the notion of $d$-embedding $\Gamma$ of a (canonical) subgeometry
$\Sigma$ and of exterior set with respect to the $h$-secant variety
$\Omega_{h}(\mathcal{A})$ of a subset $\mathcal{A}$, $ 0 \leq h \leq n-1$, in
the finite projective space $\mathrm{PG}(n-1,q^n)$, $n \geq 3$, in this article
we construct a class of non-linear $(n,n,q;d)$-MRD codes for any $ 2 \leq d
\leq n-1$. A code $\mathcal{C}_{\sigma,T}$ of this class, where $1\in T \subset
\mathbb{F}_q^*$ and $\sigma$ is a generator of
$\mathrm{Gal}(\mathbb{F}_{q^n}|\mathbb{F}_q)$, arises from a cone of
$\mathrm{PG}(n-1,q^n)$ with vertex an $(n-d-2)$-dimensional subspace over a
maximum exterior set $\mathcal{E}$ with respect to $\Omega_{d-2}(\Gamma)$. We
prove that the codes introduced in [Cossidente, A., Marino, G., Pavese, F.:
Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597--609
(2016); Durante, N., Siciliano, A.: Non-linear maximum rank distance codes in
the cyclic model for the field reduction of finite geometries. Electron. J.
Comb. (2017); Donati, G., Durante, N.: A generalization of the normal rational
curve in $\mathrm{PG}(d,q^n)$ and its associated non-linear MRD codes. Des.
Codes Cryptogr. 86, 1175--1184 (2018)] are appropriate punctured ones of
$\mathcal{C}_{\sigma,T}$ and solve completely the inequivalence issue for this
class showing that $\mathcal{C}_{\sigma,T}$ is neither equivalent nor adjointly
equivalent to the non-linear MRD code $\mathcal{C}_{n,k,\sigma,I}$, $I
\subseteq \mathbb{F}_q$, obtained in [Otal, K., \"Ozbudak, F.: Some new
non-additive maximum rank distance codes. Finite Fields and Their Applications
50, 293--303 (2018).].
