Isometries and Construction of Permutation Arrays

TL;DR

This paper explores the isometry group of permutation spaces to develop algorithms for classifying permutation codes, providing enumeration results and invariants for isometry classes.

Contribution

It introduces a new approach using isometry group characterization to classify and enumerate permutation codes up to isometry.

Findings

Numbers of non-isometric (4,3)- and (5,4)-permutation codes provided.

Constructed invariants for efficient classification.

Enumerated maximal and balanced permutation codes.

Abstract

An (n,d)-permutation code is a subset C of Sym(n) such that the Hamming distance d_H between any two distinct elements of C is at least equal to d. In this paper, we use the characterisation of the isometry group of the metric space (Sym(n),d_H) in order to develop generating algorithms with rejection of isomorphic objects. To classify the (n,d)-permutation codes up to isometry, we construct invariants and study their efficiency. We give the numbers of non-isometric (4,3)- and (5,4)- permutation codes. Maximal and balanced (n,d)-permutation codes are enumerated in a constructive way.