Advertisement

On the distance of stabilizer quantum codes from J-affine variety codes

  • Carlos Galindo
  • Olav Geil
  • Fernando Hernando
  • Diego Ruano
Article

Abstract

Self-orthogonal J-affine variety codes have been successfully used to obtain quantum stabilizer codes with excellent parameters. In a previous paper we gave formulae for the dimension of this family of quantum codes, but no bound for the minimum distance was given. In this work, we show how to derive quantum stabilizer codes with designed minimum distance from J-affine variety codes and their subfield-subcodes. Moreover, this allows us to obtain new quantum codes, some of them either with better parameters, or with larger distances than the previously known codes.

Keywords

Stabilizer J-affine variety codes Subfield-subcodes Designed minimum distance Hermitian and Euclidean duality 

References

  1. 1.
    Aly, S.A., Klappenecker, S., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53, 1183–1188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andersen, H.E., Geil, O.: Evaluation codes from order domain theory. Finite Fields Appl. 14, 92–123 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection I: statement of the problem. IEEE Trans. Inf. Theory 46, 778–788 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection II: bounds. IEEE Trans. Inf. Theory 46, 789–800 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Calderbank, A.R., Shor, P.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)ADSCrossRefGoogle Scholar
  10. 10.
    Chen, B., Ling, S., Zhang, G.: Application of constancyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61, 1474–1484 (2015)CrossRefGoogle Scholar
  11. 11.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics, 4th edn. Springer-Verlag (2015).Google Scholar
  12. 12.
    Dieks, D.: Communication by EPR devices. Phys. Rev. A 92, 271 (1982)Google Scholar
  13. 13.
  14. 14.
    Ekert, A., Macchiavello, C.: Quantum error correction for communication. Phys. Rev. Lett. 77, 2585 (1996)ADSCrossRefGoogle Scholar
  15. 15.
    Feng, K.: Quantum error correcting codes. In: Niederreiter, H. (ed) Coding Theory and Cryptology. pp. 91–142. Word Scientific. Singapore (2002)Google Scholar
  16. 16.
    Feng, K., Ma, Z.: A finite Gilbert-Varshamov bound for pure stabilizer quantum codes. IEEE Trans. Inf. Theory 50, 3323–3325 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Galindo, C., Hernando, F.: Quantum codes from affine variety codes and their subfield subcodes. Des. Codes Crytogr. 76, 89–100 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Galindo, C., Hernando, F., Ruano, D.: New quantum codes from evaluation and matrix-product codes. Finite Fields Appl. 36, 98–120 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Galindo, C., Pérez-Casales, R.: On the evaluation codes given by simple \(\delta \)-sequences. AAECC 27, 59–90 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Galindo, C., Monserrat, F.: Delta-sequences and evaluation codes defined by plane valuations at infinity. Proc. Lond. Math. Soc. 98, 714–740 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Galindo, C., Monserrat, F.: Evaluation codes defined by finite families of plane valuations at infinity. Des. Codes Crytogr. 70, 189–213 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Geil, O.: Evaluation codes from an affine variety code perspective. In: Martinez-Moro, E., Munuera, C., Ruano, D. (eds.) Advances in Algebraic Geometry Codes. Ser. Coding Theory Cryptol., vol. 5, pp. 153–180. World Scientific Publishing, Hackensack (2008)CrossRefGoogle Scholar
  24. 24.
    Geil, O.: Roots and coefficients of multivariate polynomials over finite fields. Finite Fields Appl. 23, 35–52 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Geil, O., Høholdt, T.: On hyperbolic codes. Lect. Notes Comput. Sci. 2227, 159–171 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Geil, O., Matsumoto, R., Ruano, D.: Feng–Rao decoding of primary codes. Finite Fields Appl. 34, 36–44 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Geil, O., Thomsen, C.: Weighted Reed-Muller codes revisited. Des. Codes Cryptogr. 66, 195–220 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gottesman, D.: A class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862–1868 (1996)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Grassl, M.: Bounds on the minimum distance of linear codes. http://www.codetables.de (2015). Accessed 15 Feb 2015
  30. 30.
    Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inform. 2, 757–775 (2004)CrossRefzbMATHGoogle Scholar
  31. 31.
    Grassl, M., Rötteler, M.: Quantum BCH codes. In: Proc. X Int. Symp. Theor. elec. Eng. Germany (1999), pp. 207–212Google Scholar
  32. 32.
    Hamada, M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE Trans. Inf. Theory 54, 5689–5704 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    He, X., Xu, L., Chen, H.: New \(q\)-ary quantum MDS codes with distances bigger than \(q/2\). Quantum Inf. Process. 15, 2745–2758 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Høholdt, T.: On (or in) Dick Blahut’s footprint. In: Vardy, A (ed.) Codes, Curves and Signals. The Springer International Series in Engineering and Computer Science, vol. 485, pp. 3–7 (1998). http://link.springer.com/chapter/10.1007%2F978-1-4615-5121-8_1
  35. 35.
    Jin, L., Ling, S., Luo, J., Xing, C.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE Trans. Inf. Theory 56, 4735–4740 (2010)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    La Guardia, G.G.: Construction of new families of nonbinary quantum BCH codes. Phys. Rev. A 80, 042331 (2009)ADSCrossRefGoogle Scholar
  38. 38.
    La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60, 1528–1535 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    La Guardia, G.G., Palazzo, R.: Constructions of new families of nonbinary CSS codes. Discrete Math. 310, 2935–2945 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Massey, J.L., Costello, D.J., Justensen, J.: Polynomial weights and code constructions. IEEE Trans. Inf. Theory 19, 101–110 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Matsumoto, R., Uyematsu, T.: Constructing quantum error correcting codes for \(p^m\) state systems from classical error correcting codes. IEICE Trans. Fundam. E83–A, 1878–1883 (2000)Google Scholar
  42. 42.
    Matsumoto, R., Uyematsu, T.: Lower bound for the quantum capacity of a discrete memoryless quantum channel. J. Math. Phys 43, 4391–4403 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Saints, K., Heegard, C.: On hyperbolic cascaded Reed-Solomon codes. Lect. Notes Comput. Sci. 673, 291–303 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sarvepalli, P.K., Klappenecker, A.: Nonbinary quantum Reed-Muller codes. In: Proc. 2005 Int. Symp. Information Theory, pp. 1023–1027Google Scholar
  45. 45.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In: Proc. 35th ann. symp. found. comp. sc., IEEE Comp. Soc. Press (1994), pp. 124–134Google Scholar
  46. 46.
    Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995)ADSCrossRefGoogle Scholar
  47. 47.
    Steane, A.M.: Simple quantum error correcting codes. Phys. Rev. Lett. 77, 793–797 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Steane, A.M.: Enlargement of calderbank-shor-steane quantum codes. IEEE Trans. Inf. Theory 45, 2492–2495 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)ADSCrossRefGoogle Scholar
  50. 50.
    Yu, S., Bierbrauer, J., Dong, Y., Chen, Q., Oh, C.H.: All the stabilizer codes of distance 3. IEEE Trans. Inf. Theory 59, 5179–5185 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Instituto Universitario de Matemáticas y Aplicaciones de Castellón and Departamento de MatemáticasUniversitat Jaume ICastellóSpain
  2. 2.Department of Mathematical SciencesAalborg UniversityAalborg EastDenmark

Personalised recommendations