On the minimum weight problem of permutation codes under Chebyshev distance

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 ℓ_∞-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 over 6 - ∈ for any ∈ > 0.