Non-linear MRD codes from cones over exterior sets

By using the notion of a d-embedding \(\Gamma \) $\mathrm{\Gamma }$ of a (canonical) subgeometry \(\Sigma \) $\mathrm{\Sigma }$ and of exterior sets with respect to the h-secant variety \(\Omega _{h}({\mathcal {A}})\) ${\mathrm{\Omega }}_h(\mathcal{A})$ of a subset \({\mathcal {A}}\) $\mathcal{A}$ , \( 0 \le h \le n-1\) $0\le h\le n-1$ , in the finite projective space \({\textrm{PG}}(n-1,q^n)\) $\text{PG}(n-1,q^n)$ , \(n \ge 3\) $n\ge 3$ , in this article we construct a class of non-linear (n, n, q; d)-MRD codes for any \( 2 \le d \le n-1\) $2\le d\le n-1$ . A code of this class \({\mathcal {C}}_{\sigma ,T}\) ${\mathcal{C}}_{\sigma ,T}$ , where \(1\in T \subseteq {\mathbb {F}}_q^*\) $1\in T\subseteq {\mathbb{F}}_q^{\ast }$ and \(\sigma \) $\sigma$ is a generator of \(\textrm{Gal}({\mathbb {F}}_{q^n}|{\mathbb {F}}_q)\) $\text{Gal}({\mathbb{F}}_{q^n}|{\mathbb{F}}_q)$ , arises from a cone of \({\textrm{PG}}(n-1,q^n)\) $\text{PG}(n-1,q^n)$ with vertex an \((n-d-2)\) $(n-d-2)$ -dimensional subspace over a maximum exterior set \({\mathcal {E}}\) $\mathcal{E}$ with respect to \(\Omega _{d-2}(\Gamma )\) ${\mathrm{\Omega }}_{d-2}(\mathrm{\Gamma })$ . 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 \({\mathcal {C}}_{\sigma ,T}\) ${\mathcal{C}}_{\sigma ,T}$ and we solve completely the inequivalence issue for this class showing that \({\mathcal {C}}_{\sigma ,T}\) ${\mathcal{C}}_{\sigma ,T}$ is neither equivalent nor adjointly equivalent to the non-linear MRD codes \({\mathcal {C}}_{n,k,\sigma ,I}\) ${\mathcal{C}}_{n,k,\sigma ,I}$ , \(I \subseteq {\mathbb {F}}_q\) $I\subseteq {\mathbb{F}}_q$ , obtained in Otal and Özbudak (Finite Fields Appl 50:293–303, 2018).