We apply automata theory and Karp's minimum mean weight cycle algorithm to minimum density problems in coding theory. Using this method, we find the new upper bound $53/126 \approx 0.4206$ for the minimum density of an identifying code on the infinite hexagonal grid, down from the previous record of $3/7 \approx 0.4286$.

翻译：我们应用自动机理论与卡普最小平均权重环算法，解决编码理论中的最小密度问题。利用该方法，我们找到了无限六边形网格上识别代码最小密度的新上界$53/126 \approx 0.4206$，相较于此前$3/7 \approx 0.4286$的记录有所降低。