Abstract Ovoids of the parabolic quadric Q(6, q) of $$\textrm{PG}(6,q)$$ PG ( 6 , q ) have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q(6, q), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q(6, q) two polynomials $$f_1(X,Y,Z)$$ f 1 ( X , Y , Z ) , $$f_2(X,Y,Z)$$ f 2 ( X , Y , Z ) can be associated. In this paper we classify ovoids of Q(6, q) with $$\max \{\deg (f_1),\deg (f_2)\}<(\frac{1}{6.3}q)^{\frac{3}{13}}-1$$ max { deg ( f 1 ) , deg ( f 2 ) } < ( 1 6.3 q ) 3 13 - 1 .

Abstract By 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

Abstract Let $${\mathbb K}$$ K be the Galois field $${\mathbb F}_{q^t}$$ F q t of order $$q^t, q=p^e, p$$ q t , q = p e , p a prime, $$A={{\,\mathrm{{Aut}}\,}}({\mathbb K})$$ A = Aut ( K ) be the automorphism group of $${\mathbb K}$$ K and $$\varvec{\sigma }=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d$$ σ = ( σ 0 , … , σ d - 1 ) ∈ A d , $$d \ge 1$$ d ≥ 1 . In this paper the following generalization of the Veronese map is studied: $$\begin{aligned} \nu _{d,\varvec{\sigma }} : \langle v \rangle \in {{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K}) \longrightarrow \langle v^{\sigma _0} \otimes v^{\sigma _1} \otimes \cdots \otimes v^{\sigma _{d-1}} \rangle \in {{\,\mathrm{{PG}}\,}}(n^d-1,{\mathbb K}). \end{aligned}$$ ν d , σ : ⟨ v ⟩ ∈ PG ( n - 1 , K ) ⟶ ⟨ v σ 0 ⊗ v σ 1 ⊗ ⋯ ⊗ v σ d - 1 ⟩ ∈ PG ( n d - 1 , K ) . Its image will be called the $$(d,\varvec{\sigma })$$ ( d , σ ) -Veronese variety $$\mathcal V_{d,\varvec{\sigma }}$$ V d