On large partial ovoids of symplectic and Hermitian polar spaces

In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces W ( 3 , q ) ${\mathscr{W}}(3,q)$, q $q$ odd square, q not equivalent to 0 ( mod 3 ) $q\not\equiv 0(\mathrm{mod}3)$, W ( 5 , q ) ${\mathscr{W}}(5,q)$ and of the Hermitian polar spaces Script capital H ( 4 , q 2 ) ${\rm{ {\mathcal H} }}(4,{q}<^>{2})$, q $q$ even or q $q$ odd square, q not equivalent to 0 ( mod 3 ) $q\not\equiv 0(\mathrm{mod}3)$, Script capital H ( 6 , q 2 ) ${\rm{ {\mathcal H} }}(6,{q}<^>{2})$, Script capital H ( 8 , q 2 ) ${\rm{ {\mathcal H} }}(8,{q}<^>{2})$.