  We derive simplified sphere-packing and Gilbert-Varshamov bounds for codes in
the sum-rank metric, which can be computed more efficently 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 field, almost all linear codes
achieve it.
