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. By exploiting the connection between scattered [Formula: see text]-subspaces of [Formula: see text] and minimal nondegenerate 3-dimensional rank-metric codes of [Formula: see text], [Formula: see text], described in [ G. N. Alfarano et al., J. Combin. Theory Ser. A, 192 (2022), 105658 ], we will exhibit a new class of codes with parameters [Formula: see text] for infinite values of [Formula: see text] and [Formula: see text] odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.

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

Abstract In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a junta (Combin Probab Comput 18(1–2):107–122, 2009); i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.