On optimal quantum codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [n,n - 2d + 2,d](q) exist for all 3 <= n <= q and 1 <= d <= n/2 + 1. We also present quantum MDS codes with parameters [q(2), q(2) - 2d + 2,d](q) for 1 <= d <= q which additionally give rise to shortened codes [q(2) - s, q(2) - 2d + 2 - s,d](q) for some s.