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Journal of Applied Mathematics and Computing

Erschienen in:

18.09.2019 | Original Research

Hulls of linear codes revisited with applications

verfasst von: Satanan Thipworawimon, Somphong Jitman

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2020

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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.

Vorheriger Artikel Successive iteration technique for singular nonlinear system with four-point boundary conditions

Nächster Artikel Comparative study on difference schemes for singularly perturbed boundary turning point problems with Robin boundary conditions

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

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

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

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

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

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)MATHCrossRef

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

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

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)MathSciNetMATHCrossRef

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 MathSciNetMATHCrossRef

11.

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

12.

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

13.

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

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)MathSciNetMATHCrossRef

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)MathSciNetMATHCrossRef

16.

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

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)MathSciNetMATHCrossRef

19.

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

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)MathSciNetMATHCrossRef

21.

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

22.

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

23.

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

24.

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

25.

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

26.

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

27.

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

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)CrossRef

Titel: Hulls of linear codes revisited with applications
verfasst von: Satanan Thipworawimon
Somphong Jitman
Publikationsdatum: 18.09.2019
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Applied Mathematics and Computing / Ausgabe 1-2/2020
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-019-01286-7

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