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An infinite family of non-extendable MRD codes

Daniele Bartoli, Alessandro Giannoni, Giuseppe Marino, Alessandro Neri

Abstract

In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence.

An infinite family of non-extendable MRD codes

Abstract

In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates -linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence.

Paper Structure

This paper contains 9 sections, 21 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Scattered subspaces
  4. Linearized polynomials and Dickson matrices
  5. Rank metric codes
  6. A few things on algebraic varieties
  7. An infinite family of maximally scattered subspaces in $\mathbb{F}_{q^5}^2$
  8. Self-duality of the code $\mathcal{C}_\delta$
  9. Concluding remarks and open problems

Key Result

Theorem 2.3

If $U\leq \mathbb{F}_{q^m}^k$ is a maximally scattered subspace, then

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: see MR4079480, zini2021scattered
  • Definition 2.9
  • ...and 35 more