The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches.

翻译：秩和度量提供了一个统一框架，同时推广了著名的汉明度量和秩度量，并在网络编码、分布式存储和空时编码等领域得到应用。其核心问题在于确定具有给定最小距离的码的最大尺寸。本文运用谱方法与优化技术，推导了秩和度量码尺寸的新的严格上界，其中包括一种半定规划（SDP）上界，基于计算实验，该上界能够超越现有最优上界。此外，我们比较了Delsarte线性规划（LP）上界与一种最新的特征值LP上界，并证明了二者之间的等价性，特别关注了秩和度量的极值区域。最后，我们展示了如何利用多种SDP、LP及特征值上界来证明某些最优与完美秩和度量码的不存在性结果。我们的结果表明，谱方法与优化技术的结合有效捕捉了秩和度量的混合特性，提供了克服经典编码理论方法局限性的新技术。