  Calderbank, Rains, Shor and Sloane (see \cite{Sloane}) showed that
error-correction is possible in the context of quantum computations. Quantum
stabilizer codes are a class of additive quaternary codes in binary projective
spaces, which are self-orthogonal with respect to the symplectic form. A
geometric description is given in \cite{Bierbra}, where also the notion of
quantum cap is introduced. Quantum caps correspond to the special case of
quantum stabilizer codes of distance $d=4$ when the code is linear over GF(4).
In the present paper we review the translation from quantum error-correction to
symplectic geometry and study quantum codes in PG(4,4) where we construct
complete quantum caps with 20, 29, 30, 32, 33, 34, 36 and 38 points and
incomplete quantum caps with 10, 12, 13, 20, 23, 24, 25 and 26 points and we
prove the non existence of 11-quantum caps. In particular the quantum caps of
sizes 36 and 38 yield positive answers to the existence questions of quantum
codes $[[36,26,4]]$ and $[[38,28,4]]$ that remained open in the data base
\cite{codetable}.
