Abstract Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in F q n × n the idealizers have been proved to be isomorphic to finite fields of size at most q n . Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in F q n × n for n ≤ 9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for n = 7 , q odd and for n = 8 , q ≡ 1 ( mod 3 ) . These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n ≥ 9 .

中文翻译：

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具有最大理想化器的 MRD 代码

摘要 左右理想化器是线性秩距离码的重要不变量。在 F qn × n 中的最大秩距离（简称 MRD）码的情况下，理想化器已被证明与最多为 qn 大小的有限域同构。到目前为止，唯一已知的具有最大左右理想化器的 MRD 码是广义 Gabidulin 码，该码首先由 Delsarte 于 1978 年构建，后来由 Kshevetskiy 和 Gabidulin 在 2005 年推广。本文将 MRD 码分类为 F qn × n对于具有最大左右理想化器的 n ≤ 9 并将它们连接到 Moore 型矩阵。除了广义的 Gabidulin 代码，事实证明还有一系列秩距离代码为 MRD 代码提供了 n = 7 , q 奇数和 n = 8 , q ≡ 1 ( mod 3 ) 的最大理想化器。这些代码不等同于任何先前已知的 MRD 代码。此外，我们表明这一系列秩距离代码没有为 n ≥ 9 提供任何进一步的例子。