Permutation codes over Sylow 2-subgroups $Syl_2(S_{2^n})$ of symmetric groups $S_{2^n}$

The permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be defined as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be different. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances.In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by  rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ and describe an algorithm for finding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity $O(2^n)$.     The metric properties of the codes that are defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ are studied. The capacity and number of codes for the maximum and the minimum non-trivial distance over codes are characterized.