  We derive simplified sphere-packing and Gilbert--Varshamov bounds for codes
in the sum-rank metric, which can be computed more efficiently than previous
ones. They give rise to asymptotic bounds that cover the asymptotic setting
that has not yet been considered in the literature: families of sum-rank-metric
codes whose block size grows in the code length. We also provide two genericity
results: we show that random linear codes achieve almost the sum-rank-metric
Gilbert--Varshamov bound with high probability. Furthermore, we derive bounds
on the probability that a random linear code attains the sum-rank-metric
Singleton bound, showing that for large enough extension fields, almost all
linear codes achieve it.
