In order to minimize the number of orbits that must be enumerated or
represented, and following SLEPIAN again, we can restrict attention to
indecomposable linear 
-codes. Let 
be a linear
-code over 
with generator matrix 
and let 
be a linear 
-code over with generator matrix 
,
then the code 
with generator matrix
is called the direct sum of the codes and , and it will be
denoted by 
. A code is called decomposable, if
and only if it is equivalent to a code which is the direct sum of two or more
linear codes. Otherwise it is called indecomposable.
In [13] SLEPIAN proves that every decomposable linear
-code is equivalent to a direct sum of indecomposable codes, and that
this decomposition is unique up to equivalence and order of the summands.
SLEPIAN used a generating function scheme for computing the numbers
and 
. However after constructing these codes the
authors realized that in some situations this formula doesn't work correctly.
For that reason we are giving another formula to determine the numbers
and 
. For the rest of this section let 
.
where
and where the first sum is taken over the cycle types 
.
Theorem
The number is equal to
of 
, (which means that 
and 
) such that 
, while the second sum is over the 
-tuples 
, for which 
, and
. In the same way the can be computed from the
. The numerical results show that for fixed 
and the
sequence of is unimodal and symmetric. (It is easy to prove that
this sequence must be symmetric, but the proof of the unimodality is still
open.)
Proof:
In order to evaluate the values the number of all classes of
decomposable linear -codes must be subtracted from 
. In other
words one has to find all possibilities of decomposing a linear -code
into a direct sum of indecomposable linear 
-codes such that
According to SLEPIANs theorem the -codes can be arranged
such that 
and in the case that 
the inequality 
holds. At first all partitions of into
at least two parts and into at most 
parts must be found. Let
be such a partition with 
and 
. Then the cycle type of 
is 
, where
. Two decomposable codes which
determine two different partitions of are not equivalent. In the second
step to each partition of one has to find all sequences 
such that 
is fulfilled, and such that codes belonging to
different sequences are not equivalent. In order to do this we start with
such a sequence and define
then
Two decomposable codes which on the one hand belong to one partition of
but on the other hand define two different vectors 
and 
are not in
the same equivalence class. Now the other way round we start with a sequence
and try to determine all sequences
which define non equivalent codes such that 
.
Again by SLEPIAN's theorem we must find all partitions of 
(this implies 
) into exacyly 
parts of the form
In the last step 
must be evaluated. This is the the number of
classes of linear 
-codes, which have a decomposition into a
direct sum of indecomposable 
-codes for 
to a
given partition 
of 
into exactly parts. We already know that there are
classes of indecomposable linear -codes. In the case
that all the 
are distinct, this number is given by
Otherwise there exist 
such that 
and 
. Then
and permuting the corresponding summands in the
direct sum may lead to equivalent codes. For 
let
be the cardinality of 
. Now there
is a bijection between the classes of codes, which are a direct sum of
-codes and the orbits of the symmetric group 
on the set of all mappings from into a set of 
elements,
where the action of is given by:
By PÓLYAs theorem [12] one has to compute the cycle index
and each variable must be substituted by . Doing
this for all possible values of 
we get
Using the computer algebra system SYMMETRICA, among many other ones the tables of numbers of indecomposable codes which are shown below were computed.
