On the Labeling Problem of Permutation Group Codes Under the Infinity Metric

We consider codes over permutations under the infinity norm. Given such a code, we show that a simple relabeling operation, which produces an isomorphic code, may drastically change the minimal distance of the code. Thus, we may choose a code structure for efficient encoding procedures, and then optimize the code's minimal distance via relabeling. To establish that the relabeling problem is hard and is of interest, we formally define it and show that all codes may be relabeled to get a minimal distance at most 2. On the other hand, the decision problem of whether a code may be relabeled to distance 2 or more is shown to be NP-complete, and calculating the best achievable minimal distance after relabeling is proved to be hard to approximate up to a factor of 2. We then consider general bounds on the relabeling problem. We specifically construct the optimal relabeling for transitive cyclic groups. We conclude with the main result—a general probabilistic bound, which we then use to show both the $\mathop{\rm AGL}(p)$ group and the dihedral group on $p$ elements may be relabeled to a minimal distance of $p-O(\sqrt{p\ln p})$.