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Hulls of linear codes revisited with applications

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Abstract

Hulls of linear codes have been of interest and extensively studied due to their rich algebraic structures and wide applications. In this paper, alternative characterizations of hulls of linear codes are given as well as their applications. Properties of hulls of linear codes are given in terms of their Gramians of their generator and parity-check matrices. Moreover, it is show that the Gramian of a generator matrix of every linear code over a finite field of odd characteristic is diagonalizable. Subsequently, it is shown that a linear code over a finite field of odd characteristic is complementary dual if and only if it has an orthogonal basis. Based on this characterization, constructions of good entanglement-assisted quantum error-correcting codes are provided.

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References

  1. Assmus, E.F., Key, J.D.: Affine and projective planes. Discrete Math. 83, 161–187 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brun, T., Devetak, I., Hsieh, H.M.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brun, T., Devetak, I., Hsieh, M.H.: Catalytic quantum error correction. IEEE Trans. Inf. Theory 60, 3073–3089 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10, 131–150 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over \(\mathbb{F}_q\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theory 64, 3010–3017 (2018)

    Article  MATH  Google Scholar 

  7. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86, 2605–2618 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65, 39–49 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ezerman, M.F., Jitman, S., Kiah, H.M., Ling, S.: Pure asymmetric quantum MDS codes from CSS construction: a complete characterization. Int. J. Quantum Inf. 11, 1350027 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fish, W., Key, J.D., Mwambene, E., Rodrigues, B.: Hamming graphs and special LCD codes. J. Appl. Math. Comput. (2019). https://doi.org/10.1007/s12190-019-01259-w

    Article  MathSciNet  MATH  Google Scholar 

  11. Guenda, K., Jitman, S., Gulliver, T.: A: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hsich, M.H., Devetak, I., Brun, T.: General entanglement-assisted quantum error-correcting codes. Phys. Rev. A 76, 062313 (2007)

    Article  Google Scholar 

  13. Jin, L.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63, 2843–2847 (2017)

    MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jin, L., Xing, C.: Euclidean and Hermitian self-orthogonal algebraic geometry and their application to quantum codes. IEEE Trans. Inf. Theory 58, 5484–5489 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  17. Liu, H., Pan, X.: Galois Hulls of Linear Codes Over Finite Fields, (2018) arXiv:1809.08053

  18. Leon, J.S.: Computing automorphism groups of error-correcting codes. IEEE Trans. Inf. Theory 28, 496–511 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leon, J.S.: Permutation group algorithms based on partition, I: theory and algorithms. J. Symb. Comput. 12, 533–583 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luo, G., Cao, X., Chen, X.: MDS codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inf. Theory 65, 2944–2952 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Massey, J.L.: Linear codes with complementary duals. Discrete Math. 106–107, 337–342 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pang, B., Zhu, S., Li, J.: On LCD repeated-root cyclic codes over finite fields. J. Appl. Math. Comput. 56, 625–635 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pless, V.: A classification of self-orthogonal codes over GF(2). Discrete Math. 3, 209–246 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Qian, J., Zhang, L.: Entanglement-assisted quantum codes from arbitrary binary linear codes. Des. Codes Cryptogr. 77, 193–202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sendrier, N.: On the dimension of the hull. SIAM J. Appl. Math. 10, 282–293 (1997)

    MathSciNet  MATH  Google Scholar 

  26. Sendrier, N.: Linear codes with complementary duals meet the Gilbert–Varshamov bound. Discrete Math. 285, 345–347 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sendrier, N.: Finding the permutation between equivalent codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46, 1193–1203 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sendrier, N., Skersys, G.: On the computation of the automorphism group of a linear code. In: Proceedings of IEEE ISIT’2001, p. 13. Washington, DC (2001)

  29. Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

    Satanan Thipworawimon & Somphong Jitman

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  1. Satanan Thipworawimon
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  2. Somphong Jitman
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Correspondence to Somphong Jitman.

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This research was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042.

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Thipworawimon, S., Jitman, S. Hulls of linear codes revisited with applications. J. Appl. Math. Comput. 62, 325–340 (2020). https://doi.org/10.1007/s12190-019-01286-7

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  • DOI: https://doi.org/10.1007/s12190-019-01286-7

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