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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 6/2021

06-04-2021

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

Authors: Daniele Bartoli, Maria Montanucci, Giovanni Zini

Published in: Designs, Codes and Cryptography | Issue 6/2021

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Abstract

In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdón–Bezerra–Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernández.
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Literature
1.
Abdón M., Bezerra J., Quoos L.: Further examples of maximal curves. J. Pure Appl. Algebra 213, 1192–1196 (2009).MathSciNetCrossRef
2.
Bartoli D., Montanucci M., Zini G.: AG codes and AG quantum codes from the GGS curve. Des. Codes Cryptogr. 86, 2315–2344 (2018).MathSciNetCrossRef
3.
Bartoli D., Giulietti M., Kawakita M., Montanucci M.: New examples of maximal curves with low genus. Finite Fields Appl. 68, 101744 (2020).MathSciNetCrossRef
4.
5.
Calderbank A.R., Shor P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996).CrossRef
6.
Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78(3), 405–408 (1997).MathSciNetCrossRef
7.
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, 1369–1387 (1998).MathSciNetCrossRef
8.
Chen H.: Some good quantum error-correcting codes from algebraic geometry codes. IEEE Trans. Inf. Theory 47, 2059–2061 (2001).CrossRef
9.
Feng K., Ma Z.: A finite Gilbert–Varshamov bound for pure stabilizer quantum codes. IEEE Trans. Inf. Theory 50, 3323–3325 (2004).MathSciNetCrossRef
10.
Galindo C., Hernando F.: Quantum codes from affine variety codes and their subfield-subcodes. Des. Codes Cryptogr. 76, 89–100 (2015).MathSciNetCrossRef
11.
12.
Garcia A., Güneri C., Stichtenoth H.: A generalization of the Giulietti–Korchmáros maximal curve. Adv. Geom. 10(3), 427–434 (2010).MathSciNetCrossRef
13.
Giulietti M., Korchmáros G.: A new family of maximal curves over a finite field. Math. Ann. 343(1), 229–245 (2009).MathSciNetCrossRef
14.
Goppa V.D.: Algebraic-geometric codes. Izv. Akad. Nauk SSSR Ser. Mat. 46(4), 762–781 (1982). (in Russian).MathSciNet
15.
Gottesman D.: A class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54(3), 1862–1868 (1996).MathSciNetCrossRef
16.
Grassl M., Rötteler M., Beth T.: Efficient quantum circuits for non-qubit quantum error-correcting codes. Int. J. Found. Comput. Sci. 14(5), 757–775 (2003).MathSciNetCrossRef
17.
Grassl M., Rötteler M.: Quantum MDS codes over small fields. In: IEEE International Symposium on Information Theory—Proceedings, pp. 1104–1108 (2015).
18.
Grover L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 212–219. ACM, New York (1996).
19.
Hernando F., McGuire G., Monserrat F., Moyano-Fernández J.J.: Quantum codes from a new construction of self-orthogonal algebraic geometry codes. Quantum Inf. Process. 19, 117 (2020).MathSciNetCrossRef
20.
Hirschfeld J.W.P., Korchmáros G., Torres F.: Algebraic Curves over a Finite Field. Princeton Series in Applied Mathematics, Princeton (2008).
21.
Høholdt T., van Lint J., Pellikaan R.: Algebraic geometry codes. In: Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998).
22.
Ketkar A., Klappenecker A., Kumar S., Sarvepalli P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4924 (2006).MathSciNetCrossRef
23.
Klappenecker A., Sarvepalli P.K.: Nonbinary quantum codes from Hermitian curves. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 136–143. Lecture Notes in Computer Science, vol. 3857, Springer, Berlin (2006).
24.
25.
Jin L., Xing C.P.: Euclidean and Hermitian self-orthogonal Algebraic Geometry codes and their application to Quantum codes. IEEE Trans. Inf. Theory 58, 5484–5489 (2012).MathSciNetCrossRef
26.
Kim J., Mathews G.L.: Quantum error-correcting codes from algebraic curves. In: Martinez E., Munuera C., Ruano D. (eds.) Adv. Algebraic Geom. Codes, pp. 419–444. World Scientific, Hackensack (2008).CrossRef
27.
Kim J., Walker J.: Nonbinary quantum error-correcting cods from algebraic curves. Discret. Math. 308, 3115–3124 (2008).CrossRef
28.
29.
Montanucci M., Pallozzi Lavorante V.: AG codes from the second generalization of the GK maximal curve. Discret. Math. 343(5), 101810 (2020).MathSciNetCrossRef
30.
Montanucci M., Speziali P.: The \(a\)-numbers of Fermat and Hurwitz curves. J. Pure Appl. Algebra 222(2), 477–488 (2018).MathSciNetCrossRef
31.
32.
33.
Munuera C., Tenório W., Torres F.: Quantum error-correcting codes from algebraic geometry codes of Castle Type. Quantum Inf. Process. 15, 4071–4088 (2016).MathSciNetCrossRef
34.
Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000).MATH
35.
Ouyang Y.: Permutation-invariant quantum codes. Phys. Rev. A 90(6), 062317 (2014).CrossRef
36.
Ouyang Y.: Concatenated quantum codes can attain the quantum Gilbert–Varshamov bound. IEEE Trans. Inf. Theory 60(6), 3117–3122 (2014).MathSciNetCrossRef
37.
38.
Ouyang Y., Chao R.: Permutation-invariant constant-excitation quantum codes for amplitude damping. IEEE Trans. Inf. Theory 66(5), 2921–2933 (2020).MathSciNetCrossRef
39.
Pellikaan R., Shen B.Z., van Wee G.J.M.: Which linear codes are Algebraic-Geometric. IEEE Trans. Inf. Theory 37, 583–602 (1991).MathSciNetCrossRef
40.
Ruskai M.B.: Pauli Exchange Errors in Quantum Computation. Phys. Rev. Lett. 85(1), 194–197 (2000).CrossRef
41.
Shaska T.: Quantum codes from algebraic curves with automorphisms. Condens. Matter Phys. 11, 383–396 (2008).CrossRef
42.
Shor P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM), pp. 124–134. IEEE Comput. Soc. Press, Los Alamitos (1994).
43.
Shor P.W., Smith G., Smolin J.A., Zeng B.: High performance single-error-correcting quantum codes for amplitude damping. IEEE Trans. Inf. Theory 57(10), 7180–7188 (2011).MathSciNetCrossRef
44.
Steane A.M.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. Ser. A 452, 2551–2557 (1996).MathSciNetCrossRef
45.
46.
Stichtenoth H.: Algebraic Function Fields and Codes. Graduate Texts in Mathematics, vol. 254. Springer, Berlin (2009).
47.
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).MathSciNetCrossRef
Metadata
Title
On certain self-orthogonal AG codes with applications to Quantum error-correcting codes
Authors
Daniele Bartoli
Maria Montanucci
Giovanni Zini
Publication date
06-04-2021
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 6/2021
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-021-00870-y

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