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Nonbinary quantum error-correcting codes from algebraic curves. (English) Zbl 1141.94015
The authors give a new exposition and proof of a nonbinary version of the generalized binary CSS construction. Using this construction and algebraic curves they obtain various parameters (lengths, dimensions, and minimum distances) for nonbinary quantum codes. Furthermore, they apply this construction to the tower of function fields to obtain asymptotically good nonbinary quantum codes which are constructible in polynomial time (the question of asymptotically good nonbinary quantum codes has not been considered until now).

##### MSC:
 94B60 Other types of codes 94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory 81P68 Quantum computation 14G50 Applications to coding theory and cryptography of arithmetic geometry
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##### References:
 [1] Ashikhmin, A.; Knill, E., Nonbinary quantum stabilizer codes, IEEE trans. inform. theory, 47, 3065-3072, (2001) · Zbl 1021.94033 [2] A. Ashikhmin, S. Litsyn, M.A. Tsfasman, Asymptotically good quantum codes, quant-ph/0006061, 2000. [3] H. Barnum, C. Crépeau, D. Gottesman, A. Smith, A. Tapp, Authentication of quantum messages, quant-ph/0205128, 2002. [4] Calderbank, A.R.; Rains, E.M.; Shor, P.W.; Sloane, N.J.A., Quantum error correction via codes over $$\mathit{GF}(4)$$, IEEE trans. inform. theory, 44, 1369-1387, (1998) · Zbl 0982.94029 [5] Calderbank, A.R.; Shor, P.W., Good quantum error-correcting codes exist, Phys. rev. A, 54, 1098-1105, (1996) [6] Chen, H., Some good quantum error-correcting codes from algebraic geometry codes, IEEE trans. inform. theory, 47, 2059-2061, (2001) · Zbl 1017.94028 [7] Chen, H.; Ling, S.; Xing, C., Asymptotically good quantum codes exceeding the ashikhmin – litsyn – tsfasman bound, IEEE trans. inform. theory, 47, 2055-2058, (2001) · Zbl 1017.94029 [8] I. Dumer, Concatenated codes and their multilevel generalizations, Handbook of coding theory, Vol. II, North-Holland, Amsterdam, 1998, pp. 1911-1988. · Zbl 0967.94029 [9] Feng, K., Quantum codes exist $$[[6, 2, 3]_p$$ and $$[[7, 3, 3]_p(p \geqslant 3)$$ exist, IEEE trans. inform. theory, 48, 2384-2391, (2002) · Zbl 1062.94064 [10] G.D. Forney Jr., Concatenated codes, M.I.T. Research Monograph, No. 37, The MIT Press, Cambridge, MA, 1966. [11] Garcia, A.; Stichtenoth, H., A tower of artin – schreier extensions of function fields attaining the drinfeld – vladut bound, Invent. math., 121, 211-222, (1995) · Zbl 0822.11078 [12] Goppa, V.D., Codes associated with divisors, Problemes peredachi informatsii, 13, 33-39, (1977), (English translation in Problems Inform Transmission 13 (1977) 22-27) · Zbl 0415.94005 [13] Grassl, M.; Beth, T.; Rötteler, M., On optimal quantum codes, Internat. J. quantum inform., 2, 55-64, (2004) · Zbl 1116.81012 [14] Guruswami, V.; Sudan, M., Improved decoding of reed – solomon and algebraic-geometry codes, IEEE trans. inform. theory, 45, 1757-1767, (1999) · Zbl 0958.94036 [15] Matsumoto, R., Improvement of ashikhmin – litsyn – tsfasman bound for quantum codes, IEEE trans. inform. theory, 48, 2122-2124, (2002) · Zbl 1061.94082 [16] Matsumoto, R.; Uyematsu, T., Constructing quantum error-correcting codes for $$p^m$$-state systems from classical error-correcting codes, IEICE trans. fundamentals, E83-A, 1878-1883, (2000) [17] Rains, E.M., Nonbinary quantum codes, IEEE trans. inform. theory, 45, 1827-1832, (1999) · Zbl 0958.94038 [18] Schwinger, J., Unitary operator bases, Proc. nat. acad. sci., 46, 570-579, (1960) · Zbl 0090.19006 [19] Steane, A.M., Multiple particle interference and quantum error correction, Proc. roy. soc. London ser. A, 452, 2551-2577, (1996) · Zbl 0876.94002 [20] Stichtenoth, H., Algebraic function fields and codes, (1993), Springer Berlin · Zbl 0816.14011 [21] Tsfasman, M.A.; Vladut, S.G., Algebraic – geometric codes, (1991), Kluwer Dordrecht · Zbl 0727.94007 [22] Tsfasman, M.A.; Vlaˇduţ, S.G.; Zink, Th., Modular curves, Shimura curves, and Goppa codes, better than the varshamov – gilbert bound, Math. nachr., 109, 21-28, (1982) · Zbl 0574.94013
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