Isometry classes of linear codes

A linear -code over the Galois field is a -dimensional subspace of the vector space , where denotes the set . 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 (see [8]) one can easily express equivalence of codes in terms of the wreath product action: 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 by the following definition:

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.)

As a generalization of SLEPIAN's article [11] we show in [5] that, by using LEHMANN's Lemma about actions of the wreath product, the generating function for the numbers

can be computed by the following substitution into the cycle index of the projective group acting on the -dimensional projective space :

Since the can be interpreted as numbers of classes of -matrices of rank the numbers , which are the numbers of isometry classes of linear -codes with no columns of zeros, satisfy

Restricting our attention to codes with generator matrices , such that , is injective, we can derive the numbers of isometry classes of "injective" linear codes, obtaining

where the are computed by:

In [3] it is shown how the cycle index of acting on can be computed. This paper is a generalization of [11] and of HARRISON [6][7], where the cycle indices of are computed. The idea for computing these cycle indices is the following: First determine the conjugacy classes in , which can be done by using the theory of normal forms of matrices. Then determine the number of elements in these conjugacy classes, for instance by applying a very nice formula of KUNG [9]. Finally compute the cycle type of one representative of each class. (It is well known that all elements in one conjugacy class are of the same cycle type.) These formulae have been implemented into SYMMETRICA, so a C-program for computing for can be written in the following way:

INT S_nkq_maxgrad(n,k,q,f)
OP n,k,q,f;
{
OP c,d;
INT erg=OK;
c=callocobject();
d=callocobject();
erg+=T_nkq_maxgrad(n,k,q,c);
if (gt(k,cons_eins))
{
  erg+=dec(k);
  erg+=T_nkq_maxgrad(n,k,q,d);
  erg+=inc(k);
  erg+=sub(c,d,f);
}
else erg+=copy(c,f);
erg+=freeall(c);
erg+=freeall(d);
if (erg!=OK) return error(" in computation of S_nkq_maxgrad");
return erg;
}

The program for the computation of the for is the following:

INT T_nkq_maxgrad(n,k,q,f)
OP k,q,n,f;
{
OP c,d;
INT erg=OK;
c=callocobject();
d=callocobject();
erg+=zykelind_pglkq(k,q,c);
erg+=numberofvariables(c,d);
erg+=co_polya3_sub(c,d,n,f);
erg+=freeall(c);
erg+=freeall(d);
if (erg!=OK) return error(" in computation of T_nkq_maxgrad");
return erg;
}