Quantum Information Processing, Vol. 6, No. 3, June 2007 (© 2006)
Nonbinary Quantum Goppa Codes Exceeding
the Quantum Gilbert-Varshamov Bound
Received May 6, 2006; accepted September 6, 2006; Published online: December 12, 2006
An explicit construction for nonbinary quantum Goppa codes exceeding the quan-
tum 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 imple-
mented 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
KEY WORDS: quantum error-correcting codes; coding theory; algebraic
PACS: 02.10.-v; 03.67.-a; 03.67.Pp.
Quantum error-correcting codes (QECC) have been developed in several
ways to protect quantum systems from decoherence and errors similarly
to classical coding theory. The ﬁrst ideas were published in Ref. 6. Later,
Gottesman introduced the theory of stabilizer codes.
Based on this
idea, several people developed and used methods to transform classical
codes to quantum codes in Refs. 2, 5, 7, 8, and 14. In the last years,
the focus was on binary codes but in nature a lot of nonbinary quantum
systems appear. While nonbinary quantum codes already have been con-
structed in Refs. 1, 11, and 16, nonbinary constructions using algebraic
geometry have only been considered in Ref. 13. Independently, Feng et al.
have obtained similar results in Ref. 9.
ur Mathematik und Informatik, Universit
at Mannheim, 68131 Mannheim,
Germany. E-mail: firstname.lastname@example.org
1570-0755/07/0600-0143/0 © 2006 Springer Science+Business Media, LLC