Subplanes of PG (2, q r ), Ruled Varieties V 2 r -1 2 in PG ( 2 r , q ), and Related Codes

Abstract

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