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A new method for constructing linear codes with small hulls

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Abstract

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.

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References

  1. Assmus E.F., Jr., Key J.D.: Affine and projective planes. Discret. Math. 83(2–3), 161–187 (1990).

  2. Bosma W., Cannon J.J., Fieker C., Steel A.: Handbook of Magma Functions, Edition 2.22 (2016). http://magma.maths.usyd.edu.au/magma/.

  3. Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. J. Adv. Math. Commun. 10(1), 131–150 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlet C., Li C.J., Mesnager S.: Linear codes with small hulls in semi-primitive case. Des. Codes Cryptogr. 87, 3063–3075 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  5. Carlet C., Mesnager S., Tang C.M., Qi Y.F., 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de (2019). Accessed 2 Jan 2019.

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

    MathSciNet  MATH  Google Scholar 

  10. Jin L.F., Xing C.P.: Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert–Varshamov bound. IEEE Trans. Inf. Theory 64, 6277–6282 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  11. Jitman S., Ling S.: Quasi-abelian codes. Des. Codes Cryptogr. 74, 511–531 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Li C.J., Zeng P.: Constrctions of linear codes with one-dimensional hull. IEEE Trans. Inf. Theory 65(3), 1668–1676 (2019).

    Article  MATH  Google Scholar 

  15. Lidl R., Niederreiter H., Cohn P.M.: Finite Fields. Cambridge University Press, Cambridge (1997).

    Google Scholar 

  16. Liu X.S., Liu H.L.: LCD codes over finite chain rings. Finite Fields Their Appl. 34, 1–19 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu X.S., Fan Y., Liu H.L.: Galois LCD codes over finite fields. Finite Fields Their Appl. 49, 227–242 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  18. Mesnager S., Tang C.M., Qi Y.F.: Complementary dual algebraic geometry codes. IEEE Trans. Inf. Theory 64(4), 2390–2397 (2018).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Qian L.Q., Cao X.W., Mesnager S.: Linear codes with one-dimensional hull associated with Gaussian sums. Cryptogr. Commun. 13, 225–243 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  21. Qian L.Q., Cao X.W.: On LCD and LCD double circulant codes over finite fields (submitted).

  22. 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 

  23. Sendrier N., Skersys G.: On the computation of the automorphism group of a linear code. In: Proceedings of IEEE ISIT2001, Washington, DC, p. 13 (2001).

  24. Shi M.J., Huang D.T., Sok L., Solé P.: Double circulant LCD codes over \({\mathbb{Z}}_4\). Finite Fields Their Appl. 58, 133–144 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  25. Sok L., Shi M.J., Solé P.: Constructions of optimal LCD codes over large finite fields. Finite Fields Their Appl. 50, 138–153 (2018).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors deeply thank the editor and the anonymous reviewers for their valuable comments which have highly improved the quality of the paper. This research is supported by National Natural Science Foundation of China under Grant 11771007, 12171241, 62172183 and Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant KYCX21_0175 and China Scholarship Council.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, Jiangsu, China

    Liqin Qian & Xiwang Cao

  2. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China

    Xiwang Cao

  3. School of Mathematics, Southeast University, Nanjing, 211189, Jiangsu, China

    Wei Lu

  4. I2M(CNRS, Aix-Marseille University, Centrale Marseille), Marseille, France

    Patrick Solé

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  1. Liqin Qian
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  2. Xiwang Cao
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  3. Wei Lu
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  4. Patrick Solé
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Correspondence to Xiwang Cao.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”

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Qian, L., Cao, X., Lu, W. et al. A new method for constructing linear codes with small hulls. Des. Codes Cryptogr. 90, 2663–2682 (2022). https://doi.org/10.1007/s10623-021-00940-1

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  • DOI: https://doi.org/10.1007/s10623-021-00940-1

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