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

TL;DR

This paper investigates how relabeling permutation codes under the infinity norm affects their minimal distance, revealing computational hardness and providing bounds for optimal relabeling, with implications for error correction in rank modulation.

Contribution

It formally defines the relabeling problem for permutation codes under the infinity norm and analyzes its computational complexity and bounds.

Findings

Relabeling can drastically change code distance, often reducing it to at most 2.

Deciding if a code can be relabeled to distance 1 is NP-complete.

Bounds are established for relabeling distances of cyclic groups and affine general linear groups.

Abstract

Codes over permutations under the infinity norm have been recently suggested as a coding scheme for correcting limited-magnitude errors in the rank modulation scheme. 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/decoding procedures, and then optimize the code's minimal distance via relabeling. We formally define the relabeling problem, and show that all codes may be relabeled to get a minimal distance at most 2. The decision problem of whether a code may be relabeled to distance 1 is shown to be NP-complete, and calculating the best achievable minimal distance after relabeling is proved hard to approximate. Finally, we consider general bounds on the relabeling problem. We specifically show the optimal relabeling distance of cyclic groups. A specific case of a general probabilistic argument is used to show $\agl(p)$ may be relabeled to a minimal distance of $p-O(\sqrt{p\ln p})$.