Non-Classical Hyperplanes of DW(5, q)

The hyperplanes of the symplectic dual polar space DW (5, q) arising from embedding, the so-called classical hyperplanes of DW (5, q), have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of DW (5, q). If q is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of DW (5, q). If q is odd, then we prove that every non-classical ovoid of DW (5, q) is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of DW (5, q). If DW (5, q), q odd, has a semi-singular hyperplane, then q is not a prime number.