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：我们得到简化的球体包装和吉尔伯特-瓦尔沙莫夫(Gilbert-Varshamov)的代码总标准线,可以比以前更高效地计算。它们产生无药可治的界限,覆盖文献尚未考虑的无药可治环境:区块大小在代码长度上增长的平面计量码的家属。我们还提供了两个通用结果:我们显示随机线性代码几乎达到基尔伯特-瓦尔沙莫夫(Gilbert-Varshamov)的临界值,而且概率很高。此外,我们从随机线性代码达到单顿平面编码的概率上得出界限,表明对于足够大的扩展域来说,几乎所有线性代码都达到了。