Non-linear MRD codes from cones over exterior sets

By using the notion of a d -embedding Γ Γ of a (canonical) subgeometry Σ Σ and of exterior sets with respect to the h -secant variety Ω ₕ(A) Ω h ( A ) of a subset A A , 0 ≤ h ≤ n-1 0 ≤ h ≤ n - 1 , in the finite projective space PG(n-1,qⁿ) PG ( n - 1 , q n ) , n ≥ 3 n ≥ 3 , in this article we construct a class of non-linear ( n , n , q ; d )-MRD codes for any 2 ≤ d ≤ n-1 2 ≤ d ≤ n - 1 . A code of this class Cσ ,T C σ , T , where 1∈ T ⊆ Fq^* 1 ∈ T ⊆ F q ∗ and σ σ is a generator of Gal(Fqⁿ|Fq) Gal ( F q n | F q ) , arises from a cone of PG(n-1,qⁿ) PG ( n - 1 , q n ) with vertex an $$(n-d-2)$$ ( n - d - 2 ) -dimensional subspace over a maximum exterior set E E with respect to Ω d-2(Γ ) 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 Cσ ,T C σ , T and we solve completely the inequivalence issue for this class showing that Cσ ,T C σ , T is neither equivalent nor adjointly equivalent to the non-linear MRD codes Cn,k,σ ,I C n , k , σ , I , I ⊆ Fq I ⊆ F q , obtained in Otal and Özbudak (Finite Fields Appl 50:293–303, 2018).