Subplanes of PG(2,q^r), Ruled Varieties V^2r-1₂？？？？in？PG( 2r,q), and Related Codes

In this note we consider ruled varieties $V_2^{2r？1}$ of $PG\left(2r,q\right)$ , generalizing some results shown for $r=2,3$ in previous papers. By choosing appropriately two directrix curves, a $V_2^{2r？1}$ represents a non-affine subplane of order $q$ of the projective plane $PG\left(2,q^r\right)$ represented in $PG\left(2r,q\right)$ by a spread of a hyperplane. That proves the conjecture assumed in [1]. Finally, a large family of linear codes dependent on $r\ge 2$ is associated with projective systems defined both by $V_2^{2r？1}$ and by a maximal bundle of such varieties with only an r-directrix in common, then are shown their basic parameters.