Quantum codes from a new construction of self-orthogonal algebraic geometry codes

[EN] We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.G. McGuire was partially supported by Science Foundation Ireland Grant 13/IA/1914. The remainder authors were partially supported by the Spanish Government and the EU funding program FEDER, Grants MTM2015-65764-C3-2-P and PGC2018-096446-B-C22. F. Hernando and J. J. Moyano-Fernandez are also partially supported by Universitat Jaume I, Grant UJI-B2018-10.Hernando, F.; Mcguire, G.; Monserrat Delpalillo, FJ.; Moyano-Fernández, JJ. (2020). Quantum codes from a new construction of self-orthogonal algebraic geometry codes. Quantum Information Processing. 19(4):1-25. https://doi.org/10.1007/s11128-020-2616-8S125194Abhyankar, S.S.: Irreducibility criterion for germs of analytic functions of two complex variables. Adv. Math. 74, 190–257 (1989)Abhyankar, S.S.: Algebraic Geometry for Scientists and Engineers. Mathematical Surveys and Monographs, American Mathematical Society, Providence (1990)Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection I: statement of the problem. IEEE Trans. Inf. Theory 46, 778–788 (2000)Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection II: bounds. IEEE Trans. Inf. Theory 46, 789–800 (2000)Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000)Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 76, 405–409 (1997)Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998)Campillo, A., Farrán, J.I.: Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models. Finite Fields Appl. 6, 71–92 (2000)Duursma, I.M.: Algebraic geometry codes: general theory. In: Advances in Algebraic Geometry Codes, Series of Coding Theory and Cryptology, vol. 5. World Scientific, Singapore (2008)Feng, K.: Quantum error correcting codes. In: Coding Theory and Cryptology, pp. 91–142. Word Scientific (2002)Feng, K., Ma, Z.: A finite Gilbert–Varshamov bound for pure stabilizer quantum codes. IEEE Trans. Inf. Theory 50, 3323–3325 (2004)Galindo, C., Geil, O., Hernando, F., Ruano, D.: On the distance of stabilizer quantum codes from $$J$$-affine variety codes. Quantum Inf. Process 16, 111 (2017)Galindo, C., Hernando, F., Matsumoto, R.: Quasi-cyclic construction of quantum codes. Finite Fields Appl. 52, 261–280 (2018)Galindo, C., Hernando, F., Ruano, D.: Stabilizer quantum codes from $$J$$-affine variety codes and a new Steane-like enlargement. Quantum Inf. Process 14, 3211–3231 (2015)Galindo, C., Hernando, F., Ruano, D.: Classical and quantum evaluation codes at the trace roots. IEEE Trans. Inf. Theory 16, 2593–2602 (2019)Garcia, A.: On AG codes and Artin–Schreier extensions. Commun. Algebra 20(12), 3683–3689 (1992)Goppa, V.D.: Geometry and Codes. Mathematics and its Applications, vol. 24. Kluwer, Dordrecht (1991)Goppa, V.D.: Codes associated with divisors. Probl. Inf. Transm. 13, 22–26 (1977)Gottesman, D.: A class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862–1868 (1996)Grassl, M., Rötteler, M.: Quantum BCH codes. In: Proceedings X International Symposium Theory Electrical Engineering, pp. 207–212. Germany (1999)Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2, 757–775 (2004)He, X., Xu, L., Chen, H.: New $$q$$-ary quantum MDS codes with distances bigger than $$q/2$$. Quantum Inf. Process. 15(7), 2745–2758 (2016)Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic Curves Over a Finite Field. Princeton Series in Applied Mathematics, Princeton (2008)Høholdt, T., van Lint, J., Pellikaan, R.: Algebraic geometry codes. Handb. Coding Theory 1, 871–961 (1998)Jin, L., Xing, C.: Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes. IEEE Trans. Inf. Theory 58, 4489–5484 (2012)Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4924 (2006)La Guardia, G.G.: Construction of new families of nonbinary quantum BCH codes. Phys. Rev. A 80, 042331 (2009)La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60, 1528–1535 (2014)Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)Matsumoto, R., Uyematsu, T.: Constructing quantum error correcting codes for $$p^m$$ state systems from classical error correcting codes. IEICE Trans. Fund. E83–A, 1878–1883 (2000)McGuire, G., Yılmaz, E.S.: Divisibility of L-polynomials for a family of Artin–Schreier curves. J. Pure Appl. Algebra 223(8), 3341–3358 (2019)Munuera, C., Sepúlveda, A., Torres, F.: Castle curves and codes. Adv. Math. Commun. 3, 399–408 (2009)Munuera, C., Tenório, W., Torres, F.: Quantum error-correcting codes from algebraic geometry codes of castle type. Quantum Inf. Process. 15, 4071–4088 (2016)Pellikaan, R., Shen, B.Z., van Wee, G.J.M.: Which linear codes are algebraic-geometric. IEEE Trans. Inf. Theory 37, 583–602 (1991)Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society Press (1994)Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493 (1995)Steane, A.M.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. Ser. A 452, 2551–2557 (1996)Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, Berlin (2009)Tsfasman, M.A., Vlăduţ, S.G., Zink, T.: Modular curves, Shimura curves and AG codes, better than Varshamov–Gilbert bound. Math. Nachr. 109, 21–28 (1982

Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound