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.

翻译：我们推导了和秩度量码的更简化球堆积界和吉尔伯特-瓦尔沙莫夫界，这些界比之前的计算效率更高。它们产生了渐近界，涵盖了文献中尚未考虑的渐近场景：块长随码长增长的和秩度量码族。我们还给出了两个泛化性结果：我们证明了随机线性码以高概率几乎达到和秩度量吉尔伯特-瓦尔沙莫夫界。此外，我们推导了随机线性码达到和秩度量辛格尔顿界的概率界，表明对于足够大的扩域，几乎所有线性码都能达到该界。