TITLE:
Subplanes of PG(2,qr), Ruled Varieties V2r-12inPG( 2r,q), and Related Codes
AUTHORS:
Rita Vincenti
KEYWORDS:
Finite Geometry, Translation Planes, Spreads, Varieties
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.14 No.4,
October
25,
2024
ABSTRACT: In this note we consider ruled varieties
V
2
2r−1
of
PG(
2r,q
)
, 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(
2,
q
r
)
represented in
PG(
2r,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
2r−1
and by a maximal bundle of such varieties with only an r-directrix in common, then are shown their basic parameters.