Nonbinary quantum codes

We present several results on quantum codes over general alphabets (that is , in which the fundamental units may have more than two states). In particu lar, we consider codes derived from finite symplectic geometry assumed to h ave additional global symmetries, From this standpoint, the analogs of Cald erbank-Shor-Steane codes and of GF(4)-linear codes turn out to be special c ases of the same construction. This allows us to construct families of quan tum codes from certain codes over number fields; in particular, me get anal ogs of quadratic residue codes, including a single-error-correcting code en coding one letter in five, for any alphabet size. We also consider the prob lem of fault-tolerant computation through such codes, generalizing ideas of Gottesman.