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.

翻译：特定指数在限定字段上的碎散多面体正在吸引数学中许多连接的稀有对象。 特别令人感兴趣的是第一位作者和周在2018年界定的例外对象, 其分类结果部分为已知的。 在本文中,我们建议对 $\ mathbb{F ⁇ q ⁇ n}$-线性最高距离代码进行统一的代数描述, 引入给定索引特殊线性最高级距离代码的概念。 这种连接自然会扩展一个分散的多面体在标准框架中的例外概念, 并在单项 MRD 背景下对摩尔数据集进行概括化。 我们转向特殊线性 MRD 代码的分类, 显示指数零的代码是通用的加比杜林代码, 并证明在正指数中, 该代码含有同一指数的例外分散多面体。