Exceptional scattered sequences

TL;DR

This paper introduces exceptional scattered sequences as a generalization of scattered polynomials, linking them to MRD codes, and presents new infinite families and equivalence considerations in this algebraic framework.

Contribution

It generalizes scattered polynomials to exceptional scattered sequences, establishing their connection to MRD codes and providing new infinite families and equivalence analysis.

Findings

Established the algebraic link between exceptional scattered sequences and MRD codes.

Constructed the first infinite family of exceptional scattered sequences in a nontrivial case.

Discovered a new infinite family of MRD codes as a byproduct.

Abstract

The concept of scattered polynomials is generalized to those of exceptional scattered sequences which are shown to be the natural algebraic counterpart of $\mathbb{F}_{q^n}$-linear MRD codes. The first infinite family in the first nontrivial case is also provided and equivalence issues are considered. As a byproduct, a new infinite family of MRD codes is obtained.