ABSTRACT
This chapter follows more or less the work of Johnson and Vega [812] with fewer details.
Let V be a 2d-dimensional vector space over GF (q), which admits a symplectic
form. Then there is a basis for V so that the form is
[ 0 I −I 0
] , since all forms are
conjugate. If V admits a spread of totally isotropic subspaces, we call the spread a ‘symplectic spread’ and say that we also have a ‘symplectic translation plane’. By noting that the symplectic group acts doubly transitive on totally isotropic subspaces, we see that we may choose
x = 0, y = 0, y = xM ;M t = M
as a matrix spread set (see also Biliotti, Jha, and Johnson [128]). More generally, we may allow any two components to be called x = 0, y = 0, but we shall revisit this below. We call this a ‘symmetric representation’.
