Quantum Information Processing

, Volume 14, Issue 9, pp 3211–3231 | Cite as

Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement

  • Carlos Galindo
  • Fernando Hernando
  • Diego Ruano


New stabilizer codes with parameters better than the ones available in the literature are provided in this work, in particular quantum codes with parameters \([[127,63, {\ge }12]]_2\) and \([[63,45, {\ge }6]]_4\) that are records. These codes are constructed with a new generalization of the Steane’s enlargement procedure and by considering orthogonal subfield-subcodes—with respect to the Euclidean and Hermitian inner product—of a new family of linear codes, the J-affine variety codes.


Stabilizer J-affine variety codes Subfield-subcodes  Steane enlargement Hermitian and Euclidean duality 



The authors wish to thank Ryutaroh Matsumoto and the anonymous reviewers for helpful comments on this paper.


  1. 1.
    Aly, S.A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53, 1183–1188 (2007)CrossRefGoogle Scholar
  2. 2.
    Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection II: bounds. IEEE Trans. Inf. Theory 46, 789–800 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bian, Z., et al.: Experimental determination of Ramsey numbers. Phys. Rev. Lett. 111, 130505 (2013)CrossRefADSGoogle Scholar
  6. 6.
    Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bras-Amorós, M., O’Sullivan, M.E.: Duality for some families of correction capability optimized evaluation codes. Adv. Math. Commun. 2, 15–33 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Calderbank, A.R., Shor, P.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)CrossRefADSGoogle Scholar
  11. 11.
    Delsarte, P.: On subfield subcodes of modified Reed–Solomon codes. IEEE Trans. Inf. Theory IT–21, 575–576 (1975)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dieks, D.: Communication by EPR devices. Phys. Rev. A 92, 271 (1982)Google Scholar
  13. 13.
    Ekert, A., Macchiavello, C.: Quantum error correction for communication. Phys. Rev. Lett. 77, 2585 (1996)CrossRefADSGoogle Scholar
  14. 14.
  15. 15.
    Ezerman, M.F., Jitman, S., Ling, S., Pasechnik, D.V.: CSS-like constructions of asymmetric quantum codes. IEEE Trans. Inf. Theory 59, 6732–6754 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Feng, K.: Quantum error correcting codes. In: Niederreiter H (ed.) Coding Theory and Cryptology, pp. 91–142. Word Scientific, Singapore (2002).
  17. 17.
    Feng, K., Ma, Z.: A finite Gilbert–Varshamov bound for pure stabilizer quantum codes. IEEE Trans. Inf. Theory 50, 3323–3325 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fitzgerald, J., Lax, R.F.: Decoding affine variety codes using Gröbner bases. Des. Codes Cryptogr. 13, 147–158 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Galindo, C., Hernando, F.: Quantum codes from affine variety codes and their subfield subcodes. Des. Codes Crytogr. 76, 89–100 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Galindo, C., Hernando, F., Ruano, D. New QuantumCodes from Evaluation and Matrix-Product Codes. arXiv:1406.0650
  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)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Geil, O.: Evaluation codes from an affine variety code perspective. Advances in algebraic geometry codes, Ser. Coding Theory Cryptol. 5, 153-180 (2008) World Sci. Publ., Hackensack, NJ. Eds.: E. Martinez-Moro, C. Munuera, D. RuanoGoogle Scholar
  24. 24.
    Geil, O.: Evaluation codes from order domain theory. Finite Fields Appl. 14, 92–123 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Geil, O., Matsumoto, R., Ruano, D.: Feng–Rao decoding of primary codes. Finite Fields Appl. 23, 35–52 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gottesman, D.: A class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862–1868 (1996)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    Grassl, M.: Bounds on the minimum distance of linear codes. Accessed 15th Feb 2015
  28. 28.
    Grassl, M., Rötteler, M.: Quantum BCH codes. In Proceedingss of the X International Symposium on Theoretical Electrical Engineering. Germany, pp. 207-212 1999Google Scholar
  29. 29.
    Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2, 757–775 (2004)CrossRefGoogle Scholar
  30. 30.
    Hamada, M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE Trans. Inf. Theory 54, 5689–5704 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    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
  32. 32.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    La Guardia, G.G.: Construction of new families of nonbinary quantum BCH codes. Phys. Rev. A 80, 042331 (2009)CrossRefADSGoogle Scholar
  34. 34.
    La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60, 1528–1535 (2014)CrossRefGoogle Scholar
  35. 35.
    La Guardia, G.G., Palazzo, R.: Constructions of new families of nonbinary CSS codes. Discrete Math. 310, 2935–2945 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Magma Computational Algebra System.
  37. 37.
    Marcolla, C., Orsini, E., Sala, M.: Improved decoding of affine-variety codes. J. Pure Appl. Algebra 216, 147–158 (2012)MathSciNetCrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    Matsumoto, R., Uyematsu, T.: Lower bound for the quantum capacity of a discrete memoryless quantum channel. J. Math. Phys. 43, 4391–4403 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  40. 40.
    Sarvepalli, P.K., Klappenecker, A.: Nonbinary quantum Reed–Muller codes. In: Proceedings of the 2005 International Symposium on Information Theory, pp. 1023-1027Google Scholar
  41. 41.
    Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum codes: constructions, bounds and performance. Proc. R. Soc. A 465, 1645–1672 (2000)CrossRefADSGoogle Scholar
  42. 42.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In: Proceedings of the 35th Annual Symposium on Foundations on Computer Scence. pp. 124–134, IEEE Comp. Soc. Press (1994)Google Scholar
  43. 43.
    Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995)CrossRefADSGoogle Scholar
  44. 44.
    Shor, P.W., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441–444 (2000)CrossRefADSGoogle Scholar
  45. 45.
    Smith, G., Smolin, J.: Putting, “quantumness” to the test. Physics 6, 105 (2013)CrossRefGoogle Scholar
  46. 46.
    Steane, A.M.: Simple quantum error correcting codes. Phys. Rev. Lett. 77, 793–797 (1996)zbMATHMathSciNetCrossRefADSGoogle Scholar
  47. 47.
    Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551–2577 (1996)zbMATHMathSciNetCrossRefADSGoogle Scholar
  48. 48.
    Steane, A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45, 2492–2495 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)CrossRefADSGoogle 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 2015

Authors and Affiliations

  • Carlos Galindo
    • 1
  • Fernando Hernando
    • 1
  • Diego Ruano
    • 2
  1. 1.Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones de CastellónUniversitat Jaume ICastellón de la PlanaSpain
  2. 2.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

Personalised recommendations