On the labeling problem of permutation group codes under the infinity metric

Itzhak Tamo, Moshe Schwartz

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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 resulta 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}).

Original languageEnglish
Article number6214605
Pages (from-to)6595-6604
Number of pages10
JournalIEEE Transactions on Information Theory
Issue number10
StatePublished - 26 Sep 2012


  • Error-correcting codes
  • group codes
  • permutations
  • rank modulation

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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