  Ovoids in $\PG(3, \gf(q))$ have been an interesting topic in coding theory,
combinatorics, and finite geometry for a long time. So far only two families of
ovoids are known. The first is the elliptic quadratics and the second is the
Tits ovoids. It is known that an ovoid in $\PG(3, \gf(q))$ corresponds to a
$[q^2+1, 4, q^2-q]$ code over $\gf(q)$, which is called an ovoid code. The
objectives of this paper is to study the subfield codes of the two families of
ovoid codes. The dimensions, minimum weights, and the weight distributions of
the subfield codes of the elliptic quadric codes and Tits ovoid codes are
settled. The parameters of the duals of these subfield codes are also studied.
Some of the codes presented in this paper are optimal, and some are
distance-optimal. The parameters of the subfield codes are new.
