Non-linear MRD codes from cones over exterior sets

Abstract

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