On the Existence of Agievich-Primitive Partitions
Yuriy Valerievich Tarannikov · Journal of Applied and Industrial Mathematics · 2022
We prove that for any positive integer $$ m $$ there exists a smallest positive integer $$ N=N_q(m) $$ such that for $$ n>N $$ there exist no Agievich-primitive partitions of the space $$ {\bf F}_q^n $$ 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.