Non-linear MRD codes from cones over exterior sets

TL;DR

This work presents a geometric construction of a new family of non-linear $(n,n,q;d)$-MRD codes for $2\le d\le n-1$, by leveraging cones over exterior sets and embeddings of canonical subgeometries in ${\rm PG}(n-1,q^n)$. The approach uses an $(n-k-1)$-embedding $\Gamma$ of a canonical subgeometry and a maximum exterior set $\mathcal{E}$ with respect to $\\Omega_{n-k-1}(\Gamma)$ to form a cone $\mathcal{K}(\Lambda^{\star},\mathcal{E})$, which yields MRD codes $\mathcal{C}_{\sigma,T}$ with minimum distance $d=n-k+1$, and supports puncturing relationships to known constructions. The paper shows $\mathcal{C}_{\sigma,T}$ is not equivalent to the previously known non-linear MRD codes of Otal– Özbudak in general, while highlighting special parameter regimes where equivalence occurs. It also analyzes the equivalence problem, establishing conditions under which these codes differ in structural invariants such as left idealisers, and presents a framework that connects finite geometry (exterior sets, secant varieties) to novel non-linear MRD code families. These results expand the landscape of MRD codes and link geometric configurations to coding-theoretic properties with potential applications in networks and storage systems.

Abstract

By using the notion of $d$-embedding $Γ$ of a (canonical) subgeometry $Σ$ and of exterior set with respect to the $h$-secant variety $Ω_{h}(\mathcal{A})$ of a subset $\mathcal{A}$, $ 0 \leq h \leq n-1$, in the finite projective space $\mathrm{PG}(n-1,q^n)$, $n \geq 3$, in this article we construct a class of non-linear $(n,n,q;d)$-MRD codes for any $ 2 \leq d \leq n-1$. A code $\mathcal{C}_{σ,T}$ of this class, where $1\in T \subset \mathbb{F}_q^*$ and $σ$ 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 $Ω_{d-2}(Γ)$. 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 $\mathcal{C}_{σ,T}$ and solve completely the inequivalence issue for this class showing that $\mathcal{C}_{σ,T}$ is neither equivalent nor adjointly equivalent to the non-linear MRD code $\mathcal{C}_{n,k,σ,I}$, $I \subseteq \mathbb{F}_q$, obtained in [Otal, K., Özbudak, F.: Some new non-additive maximum rank distance codes. Finite Fields and Their Applications 50, 293--303 (2018).].

Theorems & Definitions (16)

• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• Definition 5.1
• Remark 5.2
• Lemma 5.3
• Theorem 5.4
• Theorem 5.5
• proof