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 AGL(p) group and the dihedral group on p elements may be relabeled to a minimal distance of p-O(√p in p).