Linear maximum rank distance codes of exceptional type

TL;DR

This paper introduces a unified algebraic framework for exceptional linear maximum rank distance (MRD) codes, extending the concept of exceptionality from scattered polynomials to rank metric codes and advancing their classification.

Contribution

It defines exceptional linear MRD codes of a given index, generalizes Moore sets, and shows that index-zero codes are generalized Gabidulin codes, aiding classification efforts.

Findings

Index-zero codes are generalized Gabidulin codes.

Codes of positive index contain exceptional scattered polynomials.

Provides a unified algebraic description of exceptional MRD codes.

Abstract

Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of $\mathbb{F}_{q^n}$-linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.