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：[译摘要] 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$\Gamma$以及子集$\mathcal{A}$相对于$h$-截线簇$\Omega_{h}(\mathcal{A})$（$0 \leq h \leq n-1$）的外集概念，本文构造了一类对任意$2 \leq d \leq n-1$均有效的非线性$(n,n,q;d)$-MRD码。该类码中的码$\mathcal{C}_{\sigma,T}$（其中$1\in T \subset \mathbb{F}_q^*$，$\sigma$为$\mathrm{Gal}(\mathbb{F}_{q^n}|\mathbb{F}_q)$的生成元）通过对$\mathrm{PG}(n-1,q^n)$中具有$(n-d-2)$维顶点子空间的锥面构造实现，该锥面基于相对于$\Omega_{d-2}(\Gamma)$的最大外集$\mathcal{E}$。我们证明了文献[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)]中提出的码是$\mathcal{C}_{\sigma,T}$的适当穿刺码，并完全解决了该类码的非等价性问题，表明$\mathcal{C}_{\sigma,T}$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).]中基于$I \subseteq \mathbb{F}_q$构造的非线性MRD码$\mathcal{C}_{n,k,\sigma,I}$。