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：我们导出了在Sum-Rank度量中的代码的简化球装填和Gilbert-Varshamov界限，这些界限可以比以前的界限更有效地计算。它们产生了渐近界限，覆盖了文献中尚未考虑的渐近设置：Sum-Rank-Metric代码的块大小随着代码长度增长。我们还提供了两个泛性结果：我们展示随机线性代码以高概率实现几乎全部的Sum-Rank-Metric Gilbert-Varshamov界限。此外，我们导出了随机线性代码达到Sum-Rank-Metric Singleton界限的概率的界限，表明对于足够大的扩展域，几乎所有线性代码都能达到。