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.
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Niehage, A. Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound. Quantum Inf Process 6, 143–158 (2007). https://doi.org/10.1007/s11128-006-0047-9
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DOI: https://doi.org/10.1007/s11128-006-0047-9