Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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QUASI-CYCLIC SELF-DUAL CODES WITH FOUR FACTORS

  • Hyun Jin Kim (University College, Yonsei University) ;
  • Whan-Hyuk Choi (Kangwon Research Institute of Mathematical Sciences, Kangwon National University, Department of Mathematics, Kangwon National University) ;
  • Jung-Kyung Lee (College of Liberal Arts, Anyang University)
  • Received : 2024.04.24
  • Accepted : 2024.09.08
  • Published : 2024.09.30
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Abstract

In this study, we examine ℓ-quasi-cyclic self-dual codes of length ℓm over 𝔽2, provided that the polynomial Xm - 1 has exactly four distinct irreducible factors in 𝔽2[X]. We find the standard form of generator matrices of codes over the ring R ≅ 𝔽q[X]/(Xm - 1) and the conditions for the codes to be self-dual. We explicitly determine the forms of generator matrices of self-dual codes of lengths 2 and 4 over R.

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Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant founded by the Korean government(NRF-2020R1F1A1A01071645). This work was supported by 2023 Research Grant from Kangwon National University and the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2022R1C1C2011689). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korean government(NRF-2019R1G1A1099148).

References

  1. E. F. Assmus and J. D. Key, Designs and Their Codes, Cambridge: Cambridge University Press, 1992. 
  2. I. Bouyukliev, D. Bikov and S. Bouyuklieva, S-Boxes from binary quasi-cyclic codes, Electronic Notes in Discrete Mathematics 57 (2017), 67-72. https://doi.org/10.1016/j.endm.2017.02.012 
  3. J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 3rd ed., New York: Springer-Verlag, 1999. 
  4. W. Ebeling, Lattices and Codes: A Course Partially Based on Lectures by F. Hirzebruch, Advanced Lectures in Mathematics, Braunschweig: Vieweg, 1994. 
  5. M. Esmaeili, T. A. Gulliver, N. P. Secord and S. A. Mahmoud, A link between quasi-cyclic codes and convolution codes, IEEE Transactions on Information Theory 44 (1998), 431-435. https://doi.org/10.1109/18.651076 
  6. S. Han, J. L. Kim, H. Lee and Y. Lee, Construction of quasi-cyclic self-dual codes, Finite Fields and Their Applications 18 (2012), 612-633. https://doi.org/10.1016/j.ffa.2011.12.006 
  7. W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48, IEEE Transactions on Information Theory 28 (1982), 511-521. https://doi.org/10.1109/TIT.1982.1056499 
  8. T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2, IEEE Transactions on Information Theory 20 (1974), 679. https://doi.org/10.1109/TIT.1974.1055262 
  9. H. J. Kim and Y. Lee, Extremal quasi-cyclic self-dual codes over finite fields, Finite Fields and Their Applications 52 (2018), 301-318. https://doi.org/10.1016/j.ffa.2018.04.013 
  10. S. Ling and P. Sol'e, On the algebraic structure of quasi-cyclic codes I: finite fields, IEEE Transactions on Information Theory 47 (2001), 2751-2760. https://doi.org/10.1109/18.959257 
  11. S. Ling and P. Sol'e, On the algebraic structure of quasi-cyclic codes II: chain rings, Designs, Codes and Cryptography 30 (2003), 113-130. https://doi.org/10.1023/A:1024715527805 
  12. S. Ling and P. Sol'e, On the algebraic structure of quasi-cyclic codes III: generator theory, IEEE Transactions on Information Theory 51 (2005), 2692-2700. https://doi.org/10.1109/TIT.2005.850142 
  13. G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Berlin: Springer, 2006. 
  14. V. Yorgov, Binary self-dual codes with automorphism of odd order, Problems of Information Transmission 19 (1983), 260-270. 
  15. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: the user language, Journal of Symbolic Computation 24 (1997), 235-265.