This paper provides new bounds on the size of spheres in any coordinate-additive metric with a particular focus on improving existing bounds in the sum-rank metric. We derive improved upper and lower bounds based on the entropy of a distribution related to the Boltzmann distribution, which work for any coordinate-additive metric. Additionally, we derive new closed-form upper and lower bounds specifically for the sum-rank metric that outperform existing closed-form bounds.

翻译：本文提供了任意坐标加性度量中球体大小的新界，特别关注改进和秩度量中已有的界。我们基于与玻尔兹曼分布相关的分布的熵，推导了改进的上界和下界，这些界适用于任意坐标加性度量。此外，我们专门针对和秩度量推导了新的闭式上界和下界，其性能优于现有的闭式界。