## Mathematics, Department of

#### Date of this Version

7-2008

#### Citation

Published in** Discrete Math, 308 **(2008), pp 3115-3124.

doi 10.1016/j.disc.2007.08.038

#### Abstract

We give a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia- Stichtenoth tower of function fields which are constructible in polynomial time.

Binary quantum error-correcting codes have been constructed in several ways. One interesting construction uses algebraic-geometry codes [2], [6], [7], [12], with the main idea being to apply the binary CSS construction [4], [5], [16] to the asymptotically good algebraic-geometry codes arising from the Garcia-Stichtenoth [11] tower of function fields over F_{q2} (where q is a power of 2) attaining the Drinfeld-Vladut bound [17].

It is natural to consider nonbinary quantum codes. Beyond the simple fact that nonbinary error-correcting codes are interesting in the classical case, Rains [14] points out that there are indeed applications in which nonbinary quantum codes would be more appropriate than binary quantum codes. Though nonbinary quantum codes have been considered in [1], [3], [9], [14], the majority of attention has been given thus far to the binary case. In particular, the question of asymptotically good nonbinary quantum codes has not been studied until now.

## Comments

Copyright © 2007 Elsevier B.V. Used by permission.