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On $3$-dimensional MRD codes of type $\langle x^{q^t},x+δx^{q^{2t}},G(x) \rangle$

Daniele Bartoli, Francesco Ghiandoni

TL;DR

This work addresses the classification of $ ext{F}_{q^n}$-linear MRD codes of dimension $3$ by focusing on codes spanned by Moore polynomial sets of the form $igligl\langle x^{q^t}, x+ abla x^{q^{2t}}, G(x)igr angleigr floor$. The authors translate the scatteredness conditions into algebraic geometry problems via plane curves and associated varieties, and they bound intersection multiplicities at singular points to derive irreducible components not lying in the base curve. Their main contribution is a broad nonexistence result: for all $(t,q)$ outside a short list of exceptional pairs, there are no exceptional 3-dimensional $ ext{F}_{q^n}$-linear MRD codes of the proposed type with $ ext{deg}_q(G)>2t$, providing partial resolution to a 2023 conjecture. The results showcase a powerful method blending finite-field coding theory with algebraic-geometry techniques, offering a pathway to classify exceptional MRD codes and constrain Moore-set constructions in the rank-metric setting.

Abstract

In this work we present results on the classification of $\mathbb{F}_{q^n}$-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD codes $\mathcal{C}=\langle x^{q^t}, F(x), G(x) \rangle \subseteq \mathcal{L}_{n,q}$ of exceptional type, i.e. such that $\mathcal{C}$ is MRD over infinite many extensions of the field $\mathbb{F}_{q^n}$. These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.

On $3$-dimensional MRD codes of type $\langle x^{q^t},x+δx^{q^{2t}},G(x) \rangle$

TL;DR

This work addresses the classification of -linear MRD codes of dimension by focusing on codes spanned by Moore polynomial sets of the form . The authors translate the scatteredness conditions into algebraic geometry problems via plane curves and associated varieties, and they bound intersection multiplicities at singular points to derive irreducible components not lying in the base curve. Their main contribution is a broad nonexistence result: for all outside a short list of exceptional pairs, there are no exceptional 3-dimensional -linear MRD codes of the proposed type with , providing partial resolution to a 2023 conjecture. The results showcase a powerful method blending finite-field coding theory with algebraic-geometry techniques, offering a pathway to classify exceptional MRD codes and constrain Moore-set constructions in the rank-metric setting.

Abstract

In this work we present results on the classification of -linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD codes of exceptional type, i.e. such that is MRD over infinite many extensions of the field . These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.
Paper Structure (10 sections, 16 theorems, 63 equations)

This paper contains 10 sections, 16 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on algebraic curves and varieties
  3. Scattered sequences of order $1$ and MRD codes
  4. Scattered sequences and algebraic varieties
  5. Moore polynomial sets of type $\underline{f}=(x^{q^t},x+\delta x^{q^{2t}},G(x))$
  6. Study of the intersection multiplicity of branches at the singular points $P_{\xi}=(1:\xi:0),$ with $\xi \notin \mathbb{F}_{q^{\gcd(k,t)}}$
  7. Main result
  8. Case $\min\deg_q(G(X))=2t$
  9. Case $\min\deg_q(G(X))=t/2$
  10. Main theorem

Key Result

Lemma 2.1

fultonschmidt Ip eq zero o mbartoli Ip eq 0 o m Let $q$ be a prime power and $F(X,Y)\in \mathbb{F}_{q}[X,Y]$. Let $P=(\alpha,\beta)\in \mathbb{F}_{q}^{2}$ and write where $F_{i}\in \mathbb{F}_{q}[X,Y]$ is zero or homogeneous of degree $i$ and $F_{m} \neq 0.$ The following properties hold.

Theorems & Definitions (30)

  • Definition 1.1
  • Lemma 2.1
  • Remark 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • ...and 20 more