Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one tensors. Kruskal showed that the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k + d - 1$, and codes meeting this bound with equality are called minimal tensor rank (MTR) codes. It is known from algebraic complexity theory that the existence of an MTR code implies the existence of a maximum distance separable (MDS) code. In this work, we establish new results relating the tensor rank of a rank-metric code to the parameters of associated linear codes in the Hamming metric and introduce the notion of tensor rank defect. We then develop new constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes.

翻译：秩度量码是定义在有限域上的矩阵子空间，赋予秩度量，并具有自然的张量表示。张量秩衡量了将码分解为秩一张量的最小规模。Kruskal 指出，对于维数为 $k$、最小秩距离为 $d$ 的秩度量码，其张量秩至少为 $k + d - 1$，达到该下界的码称为最小张量秩（MTR）码。从代数复杂度理论可知，MTR 码的存在蕴含着最大距离可分（MDS）码的存在。本文建立了关于秩度量码张量秩与关联线性码（在汉明度量下）参数之间的新结果，并引入了张量秩亏缺概念。继而利用代数几何（AG）码，提出了具有小张量秩亏缺的秩度量码的新构造方法。