We consider the problem of deriving upper bounds on the parameters of sum-rank-metric codes, with focus on their dimension and block length. The sum-rank metric is a combination of the Hamming and the rank metric, and most of the available techniques to investigate it seem to be unable to fully capture its hybrid nature. In this paper, we introduce a new approach based on sum-rank-metric graphs, in which the vertices are tuples of matrices over a finite field, and where two such tuples are connected when their sum-rank distance is equal to one. We establish various structural properties of sum-rank-metric graphs and combine them with eigenvalue techniques to obtain bounds on the cardinality of sum-rank-metric codes. The bounds we derive improve on the best known bounds for several choices of the parameters. While our bounds are explicit only for small values of the minimum distance, they clearly indicate that spectral theory is able to capture the nature of the sum-rank-metric better than the currently available methods. They also allow us to establish new non-existence results for (possibly nonlinear) MSRD codes.

翻译：我们考虑推导和秩-度量码参数的上界问题，重点关注其维度和分组长度的上界。和秩度量是汉明度量与秩度量的组合，现有研究方法大多难以充分捕捉其混合本质。本文提出一种基于和秩度量图的新方法，其中顶点是有限域上的矩阵元组，当两个元组的和秩距离等于1时，它们之间存在边连接。我们建立了和秩度量图的各种结构性质，并将其与特征值技术相结合，从而获得和秩度量码基数上界。对于若干参数选择，我们推导的界改进了目前已知的最佳界。尽管这些界仅在最小距离较小时具有显式形式，但明确表明谱理论比现有方法更能捕捉和秩度量的本质。此外，这些界还使我们能够为（可能非线性的）MSRD码建立新的不可存在性结果。