New cubic self-dual codes of length 54, 60 and 66

Original Paper
First Online:
Received:
Accepted:
  • 108 Downloads

Abstract

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length \(3\ell \) with the algebraic approach of Ling and Solé (IEEE Trans Inf Theory 47(7):2751–2760, 2001. doi: 10.1109/18.959257). In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.

Keywords

Quasi-cyclic codes Self-dual codes Cubic construction 

Mathematics Subject Classification

94B05 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

J.-L. Kim was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF- 2016R1D1A1B03933259).

References

  1. 1.
    Bonnecaze, A., Bracco, A.D., Dougherty, S.T., Nochefranca, L.R., Solé, P.: Cubic self-dual binary codes. IEEE Trans. Inf. Theory 49(9), 2253–2259 (2003). doi: 10.1109/TIT.2003.815800 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bouyuklieva, S., Östergård, P.R.J.: New constructions of optimal self-dual binary codes of length 54. Des. Codes Cryptogr. 41(1), 101–109 (2006). doi: 10.1007/s10623-006-0018-2 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bouyuklieva, S., Yankov, N., Kim, J.L.: Classification of binary self-dual \([48,24,10]\) codes with an automorphism of odd prime order. Finite Fields Appl. 18(6), 1104–1113 (2012). doi: 10.1016/j.ffa.2012.08.002 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Conway, J.H., Sloane, N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inf. Theory 36(6), 1319–1333 (1990). doi: 10.1109/18.59931 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gulliver, T.A., Harada, M.: Weight enumerators of extremal singly-even [60, 30, 12] codes. IEEE Trans. Inf. Theory 42(2), 658–659 (1996). doi: 10.1109/18.485739 CrossRefMATHGoogle Scholar
  6. 6.
    Han, S., Kim, J.L., Lee, H., Lee, Y.: Construction of quasi-cyclic self-dual codes. Finite Fields Appl. 18(3), 613–633 (2012). doi: 10.1016/j.ffa.2011.12.006 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Harada, M., Munemasa, A.: Database of self-dual codes. http://www.math.is.tohoku.ac.jp/~munemasa/selfdualcodes.htm (2017). Accessed 09 Feb 2017
  8. 8.
    Ling, S., Solé, P.: On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inf. Theory 47(7), 2751–2760 (2001). doi: 10.1109/18.959257 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Rains, E.M., Sloane, N.J.A.: Self-dual codes. In: Pless, V.S., Huffman, W.C., Brualdi, R.A. (eds.) Handbook of Coding Theory, vol. I, II, pp. 177–294. North-Holland, Amsterdam (1998)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsMiddle East Technical UniversityÇankayaTurkey
  2. 2.Department of MathematicsSogang UniversitySeoulSouth Korea
  3. 3.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityÇankayaTurkey

Personalised recommendations