  We first present a useful characterization of additive (stabilizer) quantum
error-correcting codes. Then we present several examples of We first present a
useful characterization of additive (stabilizer) quantum error--correcting
codes. Then we present several examples of nonadditive codes. We show that
there exist infinitely many non-trivial nonadditive codes with different
minimum distances, and high rates. In fact, we show that nonadditive codes that
correct t errors can reach the asymptotic rate R=1-2H(2t/n), where H(x) is the
binary entropy function. Finally, we introduce the notion of strongly
nonadditive codes (i.e., quantum codes with the following property: the trivial
code consisting of the entire Hilbert space is the only additive code that is
equivalent to any code containing the given code), and provide a construction
for an ((11,2,3)) strongly nonadditive code.
