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.
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Grassl et al. (2004) studied this question.
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