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).].

翻译：通过利用有限射影空间$\mathrm{PG}(n-1, q^n)$ ($n \geq 3$)中(典范)子几何$\Sigma$的$d$-嵌入$\Gamma$的概念，以及子集$\mathcal{A}$相对于$h$-割线簇$\Omega_h(\mathcal{A})$ ($0 \leq h \leq n-1$)的外射集概念，本文对任意$2 \leq d \leq n-1$构造了一类非线性$(n,n,q;d)$-MRD码。该类中的码$\mathcal{C}_{\sigma, T}$（其中$1 \in T \subset \mathbb{F}_q^*$，$\sigma$是$\mathrm{Gal}(\mathbb{F}_{q^n}|\mathbb{F}_q)$的生成元）由$\mathrm{PG}(n-1,q^n)$中的锥体生成，该锥体的顶点为$(n-d-2)$维子空间，底为关于$\Omega_{d-2}(\Gamma)$的极大外射集$\mathcal{E}$。我们证明，文献[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)]中引入的码是$\mathcal{C}_{\sigma, T}$的适当穿孔码，并完全解决了该类码的非等价性问题，证明了$\mathcal{C}_{\sigma, T}$既不等价于也不伴随等价于文献[Otal, K., \"Ozbudak, F.: Some new non-additive maximum rank distance codes. Finite Fields and Their Applications 50, 293--303 (2018).]中由$\mathcal{C}_{n,k,\sigma,I}$ ($I \subseteq \mathbb{F}_q$)给出的非线性MRD码。