Nonbinary quantum goppa codes exceeding the quantum Gilbert-Varshamov bound

摘要

An explicit construction for nonbinary quantum Goppa codes exceeding the quantum Gilbert-Varshamov bound is given. First, we introduce a weighted symplectic inner product and show a method how to transform weighted codes into quantum codes with respect to the standard symplectic inner product. Then an algorithm to construct a quantum code out of any hyperelliptic curve is presented and implemented in Magma. Finally, we apply a generalization of this algorithm to a tower of function fields by Stichtenoth and show that these codes lie above the quantum Gilbert-Varshamov bound.