The sum-rank metric,
which generalizes both the Hamming and rank metric, first appeared in
2005 in the context of space-time codes and later in 2010 for multi-shot
network coding. It has since become a key tool with significant
theoretical and practical relevance.
We investigate three
classical bounds on the cardinality of sum-rank metric codes: the
Singleton, Gilbert--Varshamov, and sphere-packing bound. Although these
bounds and their asymptotic forms for the case of bounded block sizes
with a growing number of blocks are known, their evaluation relies on
computing the size of sum-rank metric balls, a task that is
super-polynomial using existing closed formulas.
To address this, we revisit
the Gilbert--Varshamov and sphere-packing bounds for linear codes,
connect them to a known polynomial-time algorithm for computing sum-rank
sphere sizes, and thereby obtain efficient polynomial-time methods for
evaluating these bounds.
Moreover, by incorporating
upper and lower estimates on sum-rank metric ball sizes, we derive
simplified variants that further reduce computational complexity. These
simplified bounds also yield new asymptotic versions of the
Gilbert--Varshamov and the sphere-packing bound that apply not only when
the number of blocks grows but also when block sizes grow and thereby
extending previously established asymptotic results.