Sum-rank codes provide a generalized framework for Hamming and rank-metric codes, with codewords represented as tuples of matrices and weight given by the sum of the block ranks. In this paper, we introduce and study a block-tensor-rank invariant for sum-rank metric codes. To each code, we associate its \emph{block tensor rank}: the smallest number of block-simple tensors, namely rank-one matrices supported inside single blocks, whose linear span contains the code. In general, determining the block tensor rank of a sum-rank code is challenging. Our main structural result shows that the block tensor rank decomposes additively across the blocks of the code, thereby reducing its computation to a tensor-rank problem on each block projection. Consequently, we derive two complementary lower bounds on the block tensor rank, referred to as the \emph{projection-wise bound} and the \emph{coordinate-code bound}. Moreover, by combining the coordinate-code bound with the classical Singleton and Griesmer bounds for codes in the Hamming metric, we obtain explicit lower bounds, called the \emph{Singleton coordinate-code bound} and the \emph{Griesmer coordinate-code bound}, respectively. We further construct families of sum-rank codes whose block tensor ranks attain the Singleton or Griesmer coordinate-code bounds. These constructions are based on Hamming-metric codes achieving the corresponding classical bounds. Finally, we show that, in certain cases, the block tensor ranks of two known families of sum-rank codes in the literature do not attain the Singleton coordinate-code bound.

翻译：和秩码为汉明码和秩度量码提供了统一的框架，其码字表示为矩阵元组，权重由各块秩之和给出。本文引入并研究了和秩度量码的块张量秩不变量。对每个码字，我们定义其块张量秩：即线性张成该码所需的最少块简单张量（即支撑在单个块内的秩一矩阵）个数。通常情况下，确定和秩码的块张量秩具有挑战性。我们的主要结构结果表明，块张量秩在码的各块之间可加性分解，从而将其计算简化为每个块投影上的张量秩问题。由此，我们推导出两个互补的块张量秩下界，分别称为投影界和坐标码界。进一步，通过将坐标码界与经典汉明度量下码的Singleton界和Griesmer界相结合，我们分别得到显式下界，即Singleton坐标码界和Griesmer坐标码界。我们还构造了块张量秩达到Singleton或Griesmer坐标码界的和秩码族，这些构造基于达到相应经典界的汉明度量码。最后，我们证明在某些情况下，文献中已知的两类和秩码族的块张量秩未达到Singleton坐标码界。