A linear 
-code over the Galois field 
is a
-dimensional subspace of the vector space 
. As usual
codewords will be written as rows 
. A 
-matrix 
over is called a generator matrix of the
linear -code 
, if and only if the rows of form a basis of
, so that 
. Two linear
-codes 
are called equivalent, if and only if there
is an isometry (with respect to the Hamming metric) which maps 
onto
. Using the notion of finite group actions one can easily express
equivalence of codes in terms of the wreath product action introduced above:
and are equivalent, if and only if there exist 
(where 
denotes the multiplicative group of the
Galois field) such that 
.
The complete monomial group 
of degree 
over
acts on 
as it was described above (see
equation (2)) in the more general case of 
on 
:
In order to apply the results of the theory of finite group actions, this
equivalence relation for linear -codes is translated into an
equivalence relation for generator matrices of linear codes, and these
generator matrices are considered to be functions 
where 
is the 
-th column of the
generator matrix . (We exclude 
-columns for obvious reasons.)
or, more explicitly,
.
Theorem
The matrices corresponding to the two functions 
and 
from to 
are generator matrices of two equivalent
codes, if and only if and lie in the same orbit of the
following action of 
as permutation group on
:
Following SLEPIAN, we use the following notation:
under the group action of 2.1, i.e.
.
under the group action of 2.1, such that
for all 
, 
and all 
the value of
is different from 
. (In the case 
, this is
the number of injective functions 
.)
-codes
over with no columns of zeros. (A linear -code has columns of
zeros, if and only if there is some 
such that 
for all
codewords 
, and so we should exclude such columns.)
-codes over with no columns of zeros. (A linear -code
is called injective, if and only if for all , and 
there is some codeword such that 
.)
-codes over with no columns of zeros. (The definition of an
indecomposable code will be given later.)
-codes over with no columns of zeros.
-codes over
with columns of zeros allowed.
As initial values we have 
for 
, 
and
for 
. It is important to realize that
is the number of orbits of functions from to
without any restrictions on the rank of the induced
matrix.
of the same
orbit have the same rank.
which induce matrices of
rank less or equal 
is 
. (This proposition holds for 
as well.)
or 
can be
computed from the 
or 
respectively, so the main
problem is the computation of the or .
In the case the wreath product becomes the group
, and so there is the group 
acting on 
and the symmetric group acting on . Applying the
formulae (3) and (4) we get
and
In the case 
the wreath product acts both on
range and domain of the functions . Applying LEHMANN's
Lemma 1.1 there is the bijection
where
and
acts on 
by 
. Using this bijection we
have to investigate the following action of 
:
where 
acts on 
by
. The set of the -orbits
is the 
-dimensional
projective space:
and the representation of as a permutation group is the projective
linear group 
.
This proves in fact the following to be true:
Thus the set of isometry classes of linear
The particular classes of elements with orders of inverse images 
.
Theorem
The isometry classes of linear -codes over are the orbits
of 
on the set of mappings 
. This set of
orbits is equal to the set of orbits of on the set 
, which can be represented by a complete set of mappings of
different content, if the content of 
is defined to be
the sequence of orders of inverse images 
.
-codes over is
equal to the set of orbits of on the set of mappings 
of different content that form -matrices of rank
.
are the classes consisting of Hamming codes.
Knowing the cycle index of acting on 
the equations
(3) and (4) can be applied again.
In [13] SLEPIAN explained how the cycle index of can
be computed using results of ELSPAS [3]. The first author
[4] generalized this concept for computing the cycle indices of
and acting on 
or respectively.
The steps of the method used were the following ones:
by applying the
theory of normal forms of matrices (or vector space endomorphisms). This
theory can be found in many textbooks of algebra.
it is important to know the exponent or subexponent of such polynomials (see
[6][11]). The exponent of such a polynomial 
is defined to be 
and the subexponent is
This element 
is uniquely defined, and it is called the integral element of 
. The exponent of can be used to compute
the cycle type of the companion or hypercompanion matrices of a monic,
irreducible polynomial , and by a direct product formula for cycle
indices the cycle types of the normal forms in 
can be derived.
Using the subexponent of and defining a formula similar to the direct
product formula of cycle indices, which depends on the integral element of
as well, the cycle type of a projectivity can be computed.
