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Open AccessOpen Access JUSTC Mathematics 16 November 2023

A representation of Galois dual codes of algebraic geometry codes via Weil differentials

  • 1.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

  • 2.

    Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2023-0019
More Information
  • Author Bio:

    Jiaqi Li is currently a postgraduate student under the supervision of Prof. Xiaowu Chen at the University of Science and Technology of China. His research mainly focuses on coding theory

    Liming Ma is a Research Associate Professor with the School of Mathematical Sciences, University of Science and Technology of China. He received his Ph.D. degree in Mathematics from Nanyang Technological University, Singapore, in 2014. From April 2014 to May 2020, he was a Lecturer with the School of Mathematical Sciences, Yangzhou University, China. His research mainly focuses on algebraic function fields over finite fields and coding theory

  • Corresponding author: E-mail: lmma20@ustc.edu.cn
  • Received Date: 13 February 2023
  • Accepted Date: 23 April 2023
    • Available Online: 16 November 2023
    Abstract

  • Abstract

    Galois dual codes are a generalization of Euclidean dual codes and Hermitian dual codes. We show that the $ h $-Galois dual code of an algebraic geometry code $ C_{ {\cal{L}},F}(D,G) $ from function field $ F/ \mathbb{F}_{p^e} $ can be represented as an algebraic geometry code $ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $ from an associated function field $ F'/ \mathbb{F}_{p^e} $ with an isomorphism $\phi_{h}:F\rightarrow F'$ satisfying $ \phi_{h}(a) = a^{p^{e-h}} $ for all $ a\in \mathbb{F}_{p^e} $. As an application of this result, we construct a family of h-Galois linear complementary dual maximum distance separable codes (h-Galois LCD MDS codes).

    Graphical abstract

    The process of representing $ {C_{{\cal L},F}}{\left( {D,G} \right)^{{ \bot _h}}} $ as an algebraic geometry code and constructing h-Galois LCD MDS codes.

    Abstract

    Galois dual codes are a generalization of Euclidean dual codes and Hermitian dual codes. We show that the $ h $-Galois dual code of an algebraic geometry code $ C_{ {\cal{L}},F}(D,G) $ from function field $ F/ \mathbb{F}_{p^e} $ can be represented as an algebraic geometry code $ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $ from an associated function field $ F'/ \mathbb{F}_{p^e} $ with an isomorphism $\phi_{h}:F\rightarrow F'$ satisfying $ \phi_{h}(a) = a^{p^{e-h}} $ for all $ a\in \mathbb{F}_{p^e} $. As an application of this result, we construct a family of h-Galois linear complementary dual maximum distance separable codes (h-Galois LCD MDS codes).

    • Public Summary

    • For any function field $ F/ \mathbb{F}_{p^e} $, there exists a function field $ F'/ \mathbb{F}_{p^e} $ with an isomorphism $ \phi_{h}:F\rightarrow F' $ satisfying $ \phi_{h}(a) = a^{p^{e-h}} $ for all $ a\in \mathbb{F}_{p^e} $.
    • We showed that the h-Galois dual code of algebraic geometry code $ C_{ {\cal{L}},F}(D,G) $ can be represented as $ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $.
    • As an application of the above result, we constructed a class of h-Galois LCD MDS codes.

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    • References(25)
  • [1]
    Goppa V D. Codes on algebraic curves. Soviet Mathematics Doklady, 1981, 24 (1): 170–172.
    [2]
    Tsfasman M A, Vlăduţ S G, Zink T. Modular curves, Shimura curves, and Goppa codes, better than the Varshamov–Gilbert bound. Mathematische Nachrichten, 1982, 109: 21–28. doi: 10.1002/mana.19821090103
    [3]
    Mesnager S, Tang C, Qi Y. Complementary dual algebraic geometry codes. IEEE Transactions on Information Theory, 2018, 64 (4): 2390–2397. doi: 10.1109/TIT.2017.2766075
    [4]
    Jin L, Kan H. Self-dual near MDS codes from elliptic curves. IEEE Transactions on Information Theory, 2019, 65 (4): 2166–2170. doi: 10.1109/TIT.2018.2880913
    [5]
    Barg A, Tamo I, Vlăduţ S. Locally recoverable codes on algebraic curves. IEEE Transactions on Information Theory, 2017, 63 (8): 4928–4939. doi: 10.1109/TIT.2017.2700859
    [6]
    Li X, Ma L, Xing C. Optimal locally repairable codes via elliptic curves. IEEE Transactions on Information Theory, 2019, 65 (1): 108–117. doi: 10.1109/TIT.2018.2844216
    [7]
    Ma L, Xing C. The group structures of automorphism groups of elliptic curves over finite fields and their applications to optimal locally repairable codes. Journal of Combinatorial Theory, Series A, 2023, 193: 105686. doi: 10.1016/j.jcta.2022.105686
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    Massey J L. Linear codes with complementary duals. Discrete Mathematics, 1992, 106–107: 337–342. doi: 10.1016/0012-365X(92)90563-U
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    Carlet C, Guilley S. Complementary dual codes for counter-measures to side-channel attacks. In: Coding Theory and Applications. Cham, Switzerland: Springer, 2015.
    [10]
    Guenda K, Jitman S, Gulliver T A. Constructions of good entanglement-assisted quantum error correcting codes. Designs, Codes and Cryptography, 2018, 86: 121–136. doi: 10.1007/s10623-017-0330-z
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    Carlet C, Mesnager S, Tang C, et al. Euclidean and Hermitian LCD MDS codes. Designs, Codes and Cryptography, 2018, 86: 2605–2618. doi: 10.1007/s10623-018-0463-8
    [12]
    Chen B, Liu H. New constructions of MDS codes with complementary duals. IEEE Transactions on Information Theory, 2018, 64 (8): 5776–5782. doi: 10.1109/TIT.2017.2748955
    [13]
    Jin L. Construction of MDS codes with complementary duals. IEEE Transactions on Information Theory, 2017, 63 (5): 2843–2847. doi: 10.1109/TIT.2016.2644660
    [14]
    Beelen P, Jin L. Explicit MDS codes with complementary duals. IEEE Transactions on Information Theory, 2018, 64 (11): 7188–7193. doi: 10.1109/TIT.2018.2816934
    [15]
    Liu H, Liu S. Construction of MDS twisted Reed–Solomon codes and LCD MDS codes. Designs, Codes and Cryptography, 2021, 89: 2051–2065. doi: 10.1007/s10623-021-00899-z
    [16]
    Shi X, Yue Q, Yang S. New LCD MDS codes constructed from generalized Reed–Solomon codes. Journal of Algebra and Its Applications, 2018, 18 (8): 1950150. doi: 10.1142/S0219498819501500
    [17]
    Fan Y, Zhang L. Galois self-dual constacyclic codes. Designs, Codes and Cryptography, 2017, 84: 473–492. doi: 10.1007/s10623-016-0282-8
    [18]
    Liu X, Fan Y, Liu H. Galois LCD codes over finite fields. Finite Fields and Their Applications, 2018, 49: 227–242. doi: 10.1016/j.ffa.2017.10.001
    [19]
    Cao M. MDS Codes with Galois hulls of arbitrary dimensions and the related entanglement-assisted quantum error correction. IEEE Transactions on Information Theory, 2021, 67 (12): 7964–7984. doi: 10.1109/TIT.2021.3117562
    [20]
    Cao M, Yang J. Intersections of linear codes and related MDS codes with new Galois hulls. arXiv: 2210.05551, 2022.
    [21]
    Fang X, Jin R, Luo J, et al. New Galois hulls of GRS codes and application to EAQECCs. Cryptography and Communications, 2022, 14: 145–159. doi: 10.1007/s12095-021-00525-8
    [22]
    Li Y, Zhu S, Li P. On MDS codes with Galois hulls of arbitrary dimensions. Cryptography and Communications, 2023, 15: 565–587. doi: 10.1007/s12095-022-00621-3
    [23]
    Wu Y, Li C, Yang S. New Galois hulls of generalized Reed–Solomon codes. Finite Fields and Their Applications, 2022, 83: 102084. doi: 10.1016/j.ffa.2022.102084
    [24]
    Stichtenoth H. Algebraic Function Fields and Codes. Berlin: Springer-Verlag, 2009.
    [25]
    Lang S. Algebra. New York: Springer-Verlag, 2002.
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    [1]
    Goppa V D. Codes on algebraic curves. Soviet Mathematics Doklady, 1981, 24 (1): 170–172.
    [2]
    Tsfasman M A, Vlăduţ S G, Zink T. Modular curves, Shimura curves, and Goppa codes, better than the Varshamov–Gilbert bound. Mathematische Nachrichten, 1982, 109: 21–28. doi: 10.1002/mana.19821090103
    [3]
    Mesnager S, Tang C, Qi Y. Complementary dual algebraic geometry codes. IEEE Transactions on Information Theory, 2018, 64 (4): 2390–2397. doi: 10.1109/TIT.2017.2766075
    [4]
    Jin L, Kan H. Self-dual near MDS codes from elliptic curves. IEEE Transactions on Information Theory, 2019, 65 (4): 2166–2170. doi: 10.1109/TIT.2018.2880913
    [5]
    Barg A, Tamo I, Vlăduţ S. Locally recoverable codes on algebraic curves. IEEE Transactions on Information Theory, 2017, 63 (8): 4928–4939. doi: 10.1109/TIT.2017.2700859
    [6]
    Li X, Ma L, Xing C. Optimal locally repairable codes via elliptic curves. IEEE Transactions on Information Theory, 2019, 65 (1): 108–117. doi: 10.1109/TIT.2018.2844216
    [7]
    Ma L, Xing C. The group structures of automorphism groups of elliptic curves over finite fields and their applications to optimal locally repairable codes. Journal of Combinatorial Theory, Series A, 2023, 193: 105686. doi: 10.1016/j.jcta.2022.105686
    [8]
    Massey J L. Linear codes with complementary duals. Discrete Mathematics, 1992, 106–107: 337–342. doi: 10.1016/0012-365X(92)90563-U
    [9]
    Carlet C, Guilley S. Complementary dual codes for counter-measures to side-channel attacks. In: Coding Theory and Applications. Cham, Switzerland: Springer, 2015.
    [10]
    Guenda K, Jitman S, Gulliver T A. Constructions of good entanglement-assisted quantum error correcting codes. Designs, Codes and Cryptography, 2018, 86: 121–136. doi: 10.1007/s10623-017-0330-z
    [11]
    Carlet C, Mesnager S, Tang C, et al. Euclidean and Hermitian LCD MDS codes. Designs, Codes and Cryptography, 2018, 86: 2605–2618. doi: 10.1007/s10623-018-0463-8
    [12]
    Chen B, Liu H. New constructions of MDS codes with complementary duals. IEEE Transactions on Information Theory, 2018, 64 (8): 5776–5782. doi: 10.1109/TIT.2017.2748955
    [13]
    Jin L. Construction of MDS codes with complementary duals. IEEE Transactions on Information Theory, 2017, 63 (5): 2843–2847. doi: 10.1109/TIT.2016.2644660
    [14]
    Beelen P, Jin L. Explicit MDS codes with complementary duals. IEEE Transactions on Information Theory, 2018, 64 (11): 7188–7193. doi: 10.1109/TIT.2018.2816934
    [15]
    Liu H, Liu S. Construction of MDS twisted Reed–Solomon codes and LCD MDS codes. Designs, Codes and Cryptography, 2021, 89: 2051–2065. doi: 10.1007/s10623-021-00899-z
    [16]
    Shi X, Yue Q, Yang S. New LCD MDS codes constructed from generalized Reed–Solomon codes. Journal of Algebra and Its Applications, 2018, 18 (8): 1950150. doi: 10.1142/S0219498819501500
    [17]
    Fan Y, Zhang L. Galois self-dual constacyclic codes. Designs, Codes and Cryptography, 2017, 84: 473–492. doi: 10.1007/s10623-016-0282-8
    [18]
    Liu X, Fan Y, Liu H. Galois LCD codes over finite fields. Finite Fields and Their Applications, 2018, 49: 227–242. doi: 10.1016/j.ffa.2017.10.001
    [19]
    Cao M. MDS Codes with Galois hulls of arbitrary dimensions and the related entanglement-assisted quantum error correction. IEEE Transactions on Information Theory, 2021, 67 (12): 7964–7984. doi: 10.1109/TIT.2021.3117562
    [20]
    Cao M, Yang J. Intersections of linear codes and related MDS codes with new Galois hulls. arXiv: 2210.05551, 2022.
    [21]
    Fang X, Jin R, Luo J, et al. New Galois hulls of GRS codes and application to EAQECCs. Cryptography and Communications, 2022, 14: 145–159. doi: 10.1007/s12095-021-00525-8
    [22]
    Li Y, Zhu S, Li P. On MDS codes with Galois hulls of arbitrary dimensions. Cryptography and Communications, 2023, 15: 565–587. doi: 10.1007/s12095-022-00621-3
    [23]
    Wu Y, Li C, Yang S. New Galois hulls of generalized Reed–Solomon codes. Finite Fields and Their Applications, 2022, 83: 102084. doi: 10.1016/j.ffa.2022.102084
    [24]
    Stichtenoth H. Algebraic Function Fields and Codes. Berlin: Springer-Verlag, 2009.
    [25]
    Lang S. Algebra. New York: Springer-Verlag, 2002.
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