On the minimum weight problem of permutation codes under Chebyshev distance

TL;DR

This paper proves that finding the minimum weight codeword in subgroup permutation codes under Chebyshev distance is NP-complete and NP-hard to approximate within a certain factor, highlighting computational complexity challenges.

Contribution

It establishes the NP-completeness and hardness of approximation for the minimum weight problem in subgroup permutation codes under Chebyshev distance.

Findings

Minimum weight codeword problem is NP-complete.

Hardness of approximation within a factor of 7/6 - ε.

Results apply to permutation codes under Chebyshev distance.

Abstract

Permutation codes of length $n$ and distance $d$ is a set of permutations on $n$ symbols, where the distance between any two elements in the set is at least $d$. Subgroup permutation codes are permutation codes with the property that the elements are closed under the operation of composition. In this paper, under the distance metric $\ell_{\infty}$-norm, we prove that finding the minimum weight codeword for subgroup permutation code is NP-complete. Moreover, we show that it is NP-hard to approximate the minimum weight within the factor $7/6-\epsilon$ for any $\epsilon>0$.