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New scattered subspaces in higher dimensions

D. Bartoli, A. Giannoni, G. Marino

Abstract

Over the past few decades, there has been extensive research on scattered subspaces, partly because of their link to MRD codes. These subspaces can be characterized using linearized polynomials over finite fields. Within this context, scattered sequences extend the concept of scattered polynomials and can be viewed as geometric equivalents of exceptional MRD codes. Up to now, only scattered sequences of orders one and two have been developed. However, this paper presents an infinite series of exceptional scattered sequences of any order beyond two which correspond to scattered subspaces that cannot be obtained as direct sum of scattered subspaces in smaller dimensions. The paper also addresses equivalence concerns within this framework.

New scattered subspaces in higher dimensions

Abstract

Over the past few decades, there has been extensive research on scattered subspaces, partly because of their link to MRD codes. These subspaces can be characterized using linearized polynomials over finite fields. Within this context, scattered sequences extend the concept of scattered polynomials and can be viewed as geometric equivalents of exceptional MRD codes. Up to now, only scattered sequences of orders one and two have been developed. However, this paper presents an infinite series of exceptional scattered sequences of any order beyond two which correspond to scattered subspaces that cannot be obtained as direct sum of scattered subspaces in smaller dimensions. The paper also addresses equivalence concerns within this framework.
Paper Structure (3 sections, 13 theorems, 84 equations)

This paper contains 3 sections, 13 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Equivalence issue

Key Result

Theorem 2.4

Assume that $\gcd(I,J)=1$ and that is not a $C_{K,m}$-power in $\mathbb{F}_{q^n}$. Then the set $U^{I,J}_{\textbf{A}}$ is scattered.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • Definition 2.7
  • ...and 22 more