Non-linear MRD codes from cones over exterior sets

Abstract

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).