Isometries and Construction of Permutation Arrays

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 characterization 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 nonisometric
(4, 3)
- and
(5, 4)
- permutation codes. Maximal and balanced
(n,d)
-permutation codes are enumerated in a constructive way.