The m-ovoids of W(5,2) and their generalizations

In this paper we are concerned with m-ovoids of the symplectic polar space W(2n+1,q), q even. In particular we show the existence of an elliptic quadric of PG(2n+1,q) not polarizing to W(2n+1,q) forming a [Formula presented]-ovoid of W(2n+1,q). A further class of (q+1)-ovoids of W(5,q) is exhibited. It arises by gluing together two orbits of a subgroup of PSp(6,q) isomorphic to PSL(2,q2). We also show that the obtained m-ovoids do not fall in any of the examples known so far in the literature. Moreover, a computer classification of the m-ovoids of W(5,2) is acquired. It turns out that W(5,2) has m-ovoids if and only if m=3 and that there are exactly three pairwise non-isomorphic examples. The first example comes from an elliptic quadric Q−(5,2) polarizing to W(5,2), whereas the other two are the 3-ovoids previously mentioned.