Partitioning of a Vector Space into Orbits by the Action of Automorphism Groups of Some Codes

Exact and asymptotic formulas are obtained for counting the number of orbits into which the space of $q^n$ vectors of length $n$ over the field $GF(q)$ is partitioned by the action of the minimal automorphism group of an arbitrary cyclic code and the automorphism group of the extended BCH code – an affine group. The calculations are carried out for $q=2$, $n=7,15,31$ and 8, 16, 32.