Linear, affine and projective groups |

Furthermore letGL_{k}(q)´GF(q)^{k}-> GF(q)^{k}(A,v) -> A·v.

The cycle indices of these two actions [2] can be computed byAFF_{k}(q)´GF(q)^{k}-> GF(q)^{k}((A,b),v) -> A·v+b.

INT zykelind_glkq(k,q,c) OP k,q,c; INT zykelind_affkq(k,q,c) OP k,q,c;In both cases

`k`

and `q`

are INTEGER objects, `k`

takes
the dimension of the vector space, `q`

the cardinality of the
finite field. The result `c`

is a POLYNOM object.
The cardinalities of *GL _{k}(q)* and

INT ordnung_glkq(k,q,c) OP k,q,c; INT ordnung_affkq(k,q,c) OP k,q,c;Again

`k`

and `q`

are INTEGER objects, `k`

takes
the dimension of the vector space, `q`

the cardinality of the
finite field. The result `c`

is an INTEGER object.
In [12] Kung gave a very elegant formula to compute the
number of matrices
in *GL _{k}(q)* which commute with a block diagonal matrix

INT kung_formel(d,lambda,q,c) OP d,lambda,q,c;computes

`c`

(an INTEGER object), which is the number of matrices,
which commute with `d`

is the degree of the
polynomial `lambda`

is the partition
`q`

is the cardinality of the
finite field. This formula can be used to determine the number of
elements in the conjugacy class of a matrix in INT number_of_irred_poly_of_degree(d,q,c) OP d,q,c;computes

`c`

(an INTEGER object) to be the number of irreducible,
monic polynomials of degree `d`

(an INTEGER object) over a finite
field with `q`

(an INTEGER object) elements.
For the enumeration of classes of linear *(n,k)*-Codes over *GF(q)*
(see
[8][6][5])
one has
to compute the cycle index of the projective group [2]
*PGL(k,q):=GF(q) ^{*}\\GL_{k}(q)* acting on the projective space

PGL(k,q)´PG(k-1,q) -> PG(k-1,q)

The cycle index of this action can be computed by(GF(q)^{*}(A),GF(q)^{*}(v)) -> GF(q)^{*}(A·v).

INT zykelind_pglkq(k,q,c) OP k,q,c;where

`k`

and `q`

are INTEGER objects as above, and `c`

is the computed cycle index a POLYNOM object.
In the case zykelind_glkq(k,q,c);zykelind_dec_apply(c);as well.

It is also possible to define linear and affine groups for
*k*-dimensional
modules over residue-class-rings *Z _{n}:=Z/nZ*. At the moment the
following two routines are just working for square free INTEGERs

INT zykelind_glkzn(k,n,c) OP k,n,c; INT zykelind_affkzn(k,n,c) OP k,n,c;In both cases

`k`

is the dimension of the module, `n`

is the
square free modulus (both are INTEGER objects) and `c`

is the
computed cycle index.
In the case *k=1* Wei and Xu computed the
cycle index of *AFF _{1}(n)* in [17] for arbitrary

INT zykelind_aff1zn(n,c) OP k,n,c;

`n`

is the modulus (an INTEGER object) and `c`

is the computed
cycle index.
harald.fripertinger@kfunigraz.ac.at,

last changed: November 19, 2001

Linear, affine and projective groups |