By using the notion of a d-embedding \(\Gamma \) of a (canonical) subgeometry \(\Sigma \) and of exterior sets with respect to the h-secant variety \(\Omega _{h}({\mathcal {A}})\) of a subset \({\mathcal {A}}\), \( 0 \le h \le n-1\), in the finite projective space \({\textrm{PG}}(n-1,q^n)\), \(n \ge 3\), in this article we construct a class of non-linear (n, n, q; d)-MRD codes for any \( 2 \le d \le n-1\). A code of this class \({\mathcal {C}}_{\sigma ,T}\), where \(1\in T \subseteq {\mathbb {F}}_q^*\) and \(\sigma \) is a generator of \(\textrm{Gal}({\mathbb {F}}_{q^n}|{\mathbb {F}}_q)\), arises from a cone of \({\textrm{PG}}(n-1,q^n)\) 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 et al (Des Codes Cryptogr 79:597–609, 2016), Donati and Durante (Des Codes Cryptogr 86:1175–1184, 2018), Durante and Siciliano (Electron J Comb, 2017) are suitable punctured ones of \({\mathcal {C}}_{\sigma ,T}\) and we 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 codes \({\mathcal {C}}_{n,k,\sigma ,I}\), \(I \subseteq {\mathbb {F}}_q\), obtained in Otal and Özbudak (Finite Fields Appl 50:293–303, 2018).

通过使用（规范）子几何\(\Sigma \ ) 的d嵌入\(\Gamma \)和相对于h割线变种\(\Omega _{h}({\ mathcal {A}})\)的子集\({\mathcal {A}}\) , \( 0 \le h \le n-1\) ，在有限射影空间\({\textrm{PG} } (n-1,q^n)\) , \ ( n \ge 3\) ，在本文中我们为任何\( 2 \le d \le n-1\) 。此类的代码\({\mathcal {C}}_{\sigma ,T}\) ，其中\(1\in T \subseteq {\mathbb {F}}_q^*\)和\(\sigma \)是\(\textrm{Gal}({\mathbb {F}}_{q^n}|{\mathbb {F}}_q)\)的生成器，由\({\textrm的圆锥体产生) {PG}}(n-1,q^n)\) 的顶点是一个\((nd-2)\)维子空间，在最大外部集合\({\mathcal {E}}\)上相对于\ (\Omega _{d-2}(\Gamma)\) 。 我们证明 Cossidente 等人 (Des Codes Cryptogr 79:597–609, 2016)、Donati 和 Durante (Des Codes Cryptogr 86:1175–1184, 2018)、Durante 和 Siciliano (Electron J Comb, 2017) 中引入的代码是合适的穿孔的\({\mathcal {C}}_{\sigma ,T}\)并且我们完全解决了此类的不等价问题，表明\({\mathcal {C}}_{\sigma ,T} \)既不等价也不伴随等价于非线性 MRD 代码\({\mathcal {C}}_{n,k,\sigma ,I}\) , \(I \subseteq {\mathbb {F}} _q\) ，在 Otal 和 Özbudak 中获得（Finite Fields Appl 50:293–303, 2018）。