Preliminaries:
Methods, Notation |

Moreover the group M_{n}(q) can be expressed as a
wreath
product of the form

M _{n}(q)=GF(q)^{*}≀S_{n}.^{.}_{.}

We use the following symbols for enumerating certain orbits
under the group action of GL_{k}(q) × M_{n}(q):

- T
_{nkq}is the number of orbits of k × n-matrices over GF(q) of rank ≤ k. - T
_{nkq}is the number of orbits of k × n-matrices over GF(q) of rank ≤ k without proportional columns. - S
_{nkq}is the number of isometry classes of (n,k)-codes over GF(q) without 0-columns (i. e. there exists no index i such that all codewords have a 0 in i-th position). - S
_{nkq}is the number of isometry classes of (n,k)-codes over GF(q) without proportional columns and without 0-columns. We call these codes*injective codes*as well. - R
_{nkq}is the number of isometry classes of indecomposable (n,k)-codes over GF(q) without 0-columns. - R
_{nkq}is the number of isometry classes of indecomposable (n,k)-codes over GF(q) without proportional columns and without 0-columns. - W
_{nkq}is the number of isometry classes of (n,k)-codes without any further requirements.

All these computations were done with the computer algebra
system SYMMETRICA:
For instance here is a short C-program which can be used for
computing the numbers T_{nkq}. The main
problem is the computation of the cycle index of the projective
linear group PGL_{k}(q) acting on PG_{k-1}(q) which
is done with the routine

zykelind_pglkq(k,q,erg) OP k,q,erg;By certain substitutions into this cycle index the desired numbers can be derived, which can be used for computing the numbers S

#include "def.h" #include "macro.h" main() { OP k,q,c,d,e,f; anfang(); k=callocobject(); q=callocobject(); c=callocobject(); d=callocobject(); e=callocobject(); f=callocobject(); printeingabe("k=? k>=2"); scan(INTEGER,k); printeingabe("q=?"); scan(INTEGER,q); zykelind_pglkq(k,q,c); numberofvariables(c,d); polya_sub(c,d,e); dec(k); zykelind_pglkq(k,q,c); numberofvariables(c,d); polya_sub(c,d,f); sub(e,f,d); println(d); freeall(k); freeall(q); freeall(c); freeall(d); freeall(e); freeall(f); ende(); }

harald.fripertinger "at" uni-graz.at, May 10, 2016

Preliminaries:
Methods, Notation |