Communication networks often rely on some form of local failover rules for fast forwarding decisions upon link failures. While on undirected networks, up to two failures can be tolerated, when just matching packet origin and destination, on directed networks tolerance to even a single failure cannot be guaranteed. Previous results have shown a lower bound of at least $\lceil\log(k+1)\rceil$ rewritable bits to tolerate $k$ failures. We improve on this lower bound for cases of $k\geq 2$, by constructing a network, in which successful routing is linked to the \textit{Covering Array Problem} on a binary alphabet, leading to a lower bound of $Ω(k + \lceil\log\log(\lceil\frac{n}{4}\rceil-k)\rceil)$ for $k$ failures in an $n$ node network.

翻译：通信网络通常依赖某种形式的本地故障转移规则，以便在链路故障时实现快速转发决策。虽然在无向网络中，仅匹配数据包源地址和目的地址即可容忍最多两次故障，但在有向网络中，甚至无法保证对单次故障的容忍。先前的研究结果表明，容忍$k$次故障至少需要$\lceil\log(k+1)\rceil$个可重写比特的下界。针对$k\geq 2$的情况，我们通过构建一个网络改进了该下界，其中成功路由与二元字母表上的\textit{覆盖阵列问题}相关联，从而推导出$n$节点网络中容忍$k$次故障的$Ω(k + \lceil\log\log(\lceil\frac{n}{4}\rceil-k)\rceil)$下界。