must there be a ground plane
Are you required to have a ground plane on the other side? Will this antenna be placed against a conducting surface when used?
Here is a type that will work if the answers to the two above questions are no.
Basically, make a vertical dipole with capacitive hat loading. To describe this, imagine a vertical dipole fed in the center running vertically across your area and in the center (left-right) of the area. Then top load the dipole with perpendicular traces at the ends of the vertical dipole. They run equal lengts to the right and left of the ends. Trim the lengths for the driving impedance to be real with no reactive component. Then use an impedance transforming network to bring the real impedance to the value you need.