Abstract
The hull of a linear code over finite fields is the intersection of the code and its dual code, which has been widely studied due to its wide applications. In this paper, we develop a general method for constructing linear codes with small hulls using the eigenvalues of the generator matrices. Using this method, we construct many optimal Euclidean and Hermitian LCD codes, which improve the previously known lower bound on the largest minimum distance. We also obtain many (near) MDS LCD codes and (near) MDS codes with one-dimensional hull. Furthermore, we give three tables about formally self-dual LCD codes.
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References
Assmus E.F. Jr., Key J.D.: Affine and projective planes. Discret. Math. 83(2–3), 161–187 (1990).
Araya M., Harada M.: On the classification of linear complementary dual codes. Discret. Math. 342(1), 270–278 (2019).
Araya M., Harada M.: On the minimum weights of binary linear complementary dual codes. Cryptogr. Commun. 12(2), 285–300 (2020).
Araya M., Harada M., Saito K.: On the minimum weights of binary LCD codes and ternary LCD codes. Finite Fields Appl. 76, 101925 (2021).
Bhargava M., Zieve M.E.: Factoring Dickson polynomials over finite fields. Finite Fields Appl. 5(2), 103–111 (1999).
Bosma W., Cannon J., Playoust C.: The Magma algebra system I: the user language. J. Symbol. Comput. 24, 235–265 (1997).
Bouyuklieva S.: Optimal binary LCD codes. Des. Codes Cryptogr. 89(11), 2445–2461 (2021).
Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10(1), 131–150 (2016).
Carlet C., Güneri C., Özbudak F., Solé P.: A new concatenated type construction for LCD codes and isometry codes. Discret. Math. 341(3), 830–835 (2018).
Carlet C., Li C., Mesnager S.: Linear codes with small hulls in semi-primitive case. Des. Codes Cryptogr. 87(12), 3063–3075 (2019).
Carlet C., Mesnager S., Tang C., Qi Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65(1), 39–49 (2019).
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(4), 3010–3017 (2018).
Dougherty S.T., Kim J., Özkaya B., Sok L., Solé P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4(2–3), 116–128 (2017).
Fu Q., Li R., Fu F., Rao Y.: On the construction of binary optimal LCD codes with short length. Int. J. Found. Comput. Sci. 30, 1237–1245 (2019).
Galvez L., Kim J.L., Lee N., Roe Y.G., Won B.S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10(4), 719–728 (2018).
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de. Accessed 4 (2021).
Güneri C., Özkaya B., Solé P.: Quasi-cyclic complementary dual codes. Finite Fields Appl. 42, 67–80 (2016).
Harada M.: Some optimal entanglement-assisted quantum codes constructed from quaternary Hermitian linear complementary dual codes. Int. J. Quant. Inf. 17, 1950053 (2019).
Harada M., Saito K.: Binary linear complementary dual codes. Cryptogr. Commun. 11(4), 677–696 (2019).
Kennedy G.T., Pless V.: On designs and formally self-dual codes. Des. Codes Cryptogr. 4(1), 43–55 (1994).
Lai C., Ashikhmin A.: Linear programming bounds for entanglement-assisted quantum error-correcting codes by split weight enumerators. IEEE Trans. Inf. Theory 64(1), 622–639 (2018).
Leon J.: Permutation group algorithms based on partition I: theory and algorithms. J. Symb. Comput. 12(4–5), 533–583 (1982).
Li C., Zeng P.: Constructions of linear codes with one-dimensional hull. IEEE Trans. Inf. Theory 65(3), 1668–1676 (2019).
Liu Z., Wang J.: Further results on Euclidean and Hermitian linear complementary dual codes. Finite Fields Appl. 59, 104–133 (2019).
Lu L., Zhan X., Yang S., Cao H.: Optimal quaternary Hermitian LCD codes. https://arxiv.org/pdf/2010.10166.pdf.
Massey J.: Linear codes with complementary duals. Discret. Math. 106–107, 337–342 (1992).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes, Amsterdam. North-Holland, The Netherlands (1977).
Qian L., Cao X., Mesnager S.: Linear codes with one-dimensional hull associated with Gaussian sums. Cryptogr. Commun. 13(2), 225–243 (2021).
Qian L., Cao X., Lu W., Solé P.: A new method for constructing linear codes with small hulls. Des. Codes Cryptogr. 90(11), 2663–2682 (2022).
Qian L., Shi M., Solé P.: On self-dual and LCD quasi-twisted codes of index two over a special chain ring. Cryptogr. Commun. 11(4), 717–734 (2019).
Sendrier N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discret. Math. 285(1), 345–347 (2004).
Sendrier N.: Finding the permutation between equivalent codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46(4), 1193–1203 (2000).
Sendrier N., Skersys G.: On the computation of the automorphism group of a linear code. In: Proc. IEEE Int. Symp. Inf. Theory, Washington, DC, p. 13 (2001).
Shi M., Huang D., Sok L., Solé P.: Double circulant LCD codes over \({\mathbb{Z} }_4\). Finite Fields Appl. 58, 133–144 (2019).
Shi M., Huang D., Sok L., Solé P.: Double circulant self-dual and LCD codes over Galois rings. Adv. Math. Commun. 13(1), 171–183 (2019).
Shi M., Li S., Kim J., Solé P.: LCD and ACD codes over a noncommutative non-unital ring with four elements. Cryptogr. Commun. 14(3), 627–640 (2022).
Shi M., Özbudak F., Xu L., Solé P.: LCD codes from tridiagonal Toeplitz matrices. Finite Fields Appl. 75, 101892 (2021).
Shi M., Qian L., Solé P.: On self-dual negacirculant codes of index two and four. Des. Codes Cryptogr. 86(11), 2485–2494 (2018).
Shi M., Xu L., Solé P.: Construction of isodual codes from polycirculant matrices. Des. Codes Cryptogr. 88(12), 2547–2560 (2020).
Shi M., Xu L., Solé P.: On isodual double Toeplitz codes. https://arxiv.org/abs/2102.09233.pdf.
Shi M., Zhu H., Qian L., Sok L., Solé P.: On self-dual and LCD double circulant and double negacirculant codes over \({\mathbb{F} }_q+u{\mathbb{F} }_q\). Cryptogr. Commun. 12, 53–70 (2020).
Sok L., Shi M., Solé P.: Construction of optimal LCD codes over large finite fields. Finite Fields Appl. 50(1), 138–153 (2018).
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The authors would also like to thank the editor and the anonymous referees for helpful comments which have highly improved the quality of the paper.
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This research is supported by the National Natural Science Foundation of China (12071001) and 2021 University Graduate Research Project (Y020410077).
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Li, S., Shi, M. & Wang, J. An improved method for constructing formally self-dual codes with small hulls. Des. Codes Cryptogr. 91, 2563–2583 (2023). https://doi.org/10.1007/s10623-023-01210-y
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DOI: https://doi.org/10.1007/s10623-023-01210-y