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

V.A Olshevska

doi:10.15421/242107

Abstract

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.

Highlights

A permutation codes is studied since the 1970s
The permutation code of length n and with minimum distance d over metric d is the set of permutations C ∈ Sn such that for every pair of dierent permutations π, σ ∈ C the distance between π and σ is greater or equal to d
Permutation codes are used as error-correction codes in channels with low power-line communication

Summary

Introduction

The permutation code of length n and with minimum distance d over metric d is the set of permutations C ∈ Sn such that for every pair of dierent permutations π, σ ∈ C the distance between π and σ is greater or equal to d. It is considered Hamming distance between permutations. We discuss properties of codes in the case when permutation codes are dened on Sylow. We study the property of the Hamming distance dH dened on permutations from Sylow 2-subgroup Syl (S2n ) of symmetric group S2n and describe an algorithm for nding the Hamming distance over elements from. Sylow 2-subgroup of the symmetric group with complexity O(2n )

Preliminaries

Hamming distance between two permutations from

Hamming distance search algorithm

Permutation codes over Sylow 2-subgroup of symmetric group

Hamming distance properties

Permutation codes over