Tensor codes are a generalisation of matrix codes. Such codes are defined as subspaces of order-r tensors for which the ambient space is endowed with the tensor-rank as a metric. A class of these codes was introduced by Roth, who also outlined a decoding algorithm for low tensor-rank errors that can be generalised to an algorithm with exponential complexity in the decoding radius. They may be viewed as a generalisation of the well-known Delsarte-Gabidulin-Roth maximum rank distance codes. We study a generalised class of these codes. We investigate their properties and outline decoding techniques for different metrics that leverage their tensor structure. We first consider a fibre-wise decoding approach, as each fibre of a codeword corresponds to a Gabidulin codeword. We then give a generalisation of Loidreau-Overbeck's decoding method that corrects errors with properties constrained by the dimensions of the slice spaces and fibre spaces. The metrics we consider are bounded from above by the tensor-rank metric, and therefore these algorithms also decode tensor-rank weight errors.

翻译：张量码是矩阵码的推广。此类码定义为阶为r的张量子空间，其背景空间以张量秩为度量。Roth曾引入一类此类码，并概述了一种针对低张量秩错误的解码算法，该算法可推广为解码半径内指数复杂度的算法。这类码可视为著名的Delsarte-Gabidulin-Roth最大秩距离码的推广。我们研究此类码的推广形式，探究其性质，并概述针对不同度量、利用其张量结构的解码技术。首先考虑纤维式解码方法，因为码字的每个纤维对应一个Gabidulin码字。随后给出Loidreau-Overbeck解码方法的推广，该算法可纠正错误，且错误特性受切片空间与纤维空间维数约束。所考虑的度量以张量秩度量为上界，因此这些算法也能解码张量秩权重错误。