  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.
