Non-linear MRD codes from cones over exterior sets

Abstract

By using the notion of $d$-embedding $\hat{I}“$ of a (canonical) subgeometry $\hat{I}\pounds$ and of exterior set with respect to the $h$-secant variety $\hat{I}{\text{c◯}}_h(\mathcal{A})$ of a subset $\mathcal{A}$, $0\le h\le n-1$, in the finite projective space $\mathrm{PG}(n-1,q^n)$, $n\ge 3$, 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 ${\mathcal{C}}_{\ddot{I}ƒ,T}$ of this class, where $1\in T\subset {\mathbb{F}}_q^{\ast }$ and $\ddot{I}ƒ$ is a generator of $\mathrm{Gal}({\mathbb{F}}_{q^n}|{\mathbb{F}}_q)$, arises from a cone of $\mathrm{PG}(n-1,q^n)$ with vertex an $(n-d-2)$-dimensional subspace over a maximum exterior set $\mathcal{E}$ with respect to $\hat{I}{\text{c◯}}_{d-2}(\hat{I}“)$. We prove that the codes introduced in [Cossidente, A., Marino, G., Pavese, F.: Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597--609 (2016); Durante, N., Siciliano, A.: Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries. Electron. J. Comb. (2017); Donati, G., Durante, N.: A generalization of the normal rational curve in $\mathrm{PG}(d,q^n)$ and its associated non-linear MRD codes. Des. Codes Cryptogr. 86, 1175--1184 (2018)] are appropriate punctured ones of ${\mathcal{C}}_{\ddot{I}ƒ,T}$ and solve completely the inequivalence issue for this class showing that ${\mathcal{C}}_{\ddot{I}ƒ,T}$ is neither equivalent nor adjointly equivalent to the non-linear MRD code ${\mathcal{C}}_{n,k,\ddot{I}ƒ,I}$, $I\subseteq {\mathbb{F}}_q$, obtained in [Otal, K., Özbudak, F.: Some new non-additive maximum rank distance codes. Finite Fields and Their Applications 50, 293--303 (2018).].