On the existence of Agievich-primitive partitions

We prove that for any positive integer $m$ there exists the smallest positive integer $N=N_q(m)$ such that for $n>N$ there are no Agievich-primitive partitions of the space $\mathbf{F}_q^n$ into $q^m$ affine subspaces of dimension $n-m$. We give lower and upper bounds on the value $N_q(m)$ and prove that $N_q(2)=q+1$. Results of the same type for partitions into coordinate subspaces are established. Bibliogr. 16.