Skip to main content
SpringerLink
Log in

Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by 12 and quaternary near-extremal Hermitian self-dual codes of length divisible by 6. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length 12m for \(m =3,4,5,6\). Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length 6m for \(m =4,5,6\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availibility

Available from the corresponding author on reasonable request

Code availability

Not applicable.

References

  1. Araya M., Harada, M.: http://yuki.cs.inf.shizuoka.ac.jp/WE/ (2022). Accessed 2 Dec 2022.

  2. Assmus E.F. Jr., Mattson H.F. Jr.: New 5-designs. J. Comb. Theory 6, 122–151 (1969).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Conway J.H., Pless V.: Monomials of orders 7 and 11 cannot be in the group of a \((24,12,10)\) self-dual quaternary code. IEEE Trans. Inform. Theory 29, 137–140 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  5. Conway J.H., Sloane N.J.A.: Sphere Packing, Lattices and Groups, 3rd edn Springer, New York (1999).

    Book  Google Scholar 

  6. Conway J.H., Pless V., Sloane N.J.A.: Self-dual codes over \(GF(3)\) and \(GF(4)\) of length not exceeding \(16\). IEEE Trans. Inform. Theory 25, 312–322 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  7. Grassl M., Gulliver T.A.: On circulant self-dual codes over small fields. Des. Codes Cryptogr. 52, 57–81 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  8. Gulliver T.A.: Optimal double circulant self-dual codes over \(\mathbb{F} _4\). IEEE Trans. Inform. Theory 46, 271–274 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  9. Gulliver T.A., Harada M.: New nonbinary self-dual codes. IEEE Trans. Inform. Theory 54, 415–417 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  10. Gulliver T.A., Harada M., Miyabayashi H.: Optimal double circulant self-dual codes over \(\mathbb{F} _4\) II. Australas. J. Comb. 39, 163–174 (2007).

    MATH  Google Scholar 

  11. Han S., Kim J.-L.: The nonexistence of near-extremal formally self-dual codes. Des. Codes Cryptogr. 51, 69–77 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  12. Harada M., Munemasa A.: A complete classification of ternary self-dual codes of length \(24\). J. Comb. Theory Ser. A 116, 1063–1072 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  13. Harada M., Munemasa A.: Classification of quaternary Hermitian self-dual codes of length \(20\). IEEE Trans. Inform. Theory 57, 3758–3762 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  14. Harada M., Holzmann W., Kharaghani H., Khorvash M.: Extremal ternary self-dual codes constructed from negacirculant matrices. Gr. Comb. 23, 401–417 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  15. Harada M., Lam C., Munemasa A., Tonchev V.D.: Classification of generalized Hadamard matrices \(H(6,3)\) and quaternary Hermitian self-dual codes of length \(18\). Electron. J. Comb. 17, 14 (2010).

    MathSciNet  MATH  Google Scholar 

  16. Kim J.-L.: New self-dual codes over \({\rm GF}(4)\) with the highest known minimum weights. IEEE Trans. Inform. Theory 47, 1575–1580 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  17. Lam C.W.H., Pless V.: There is no \((24,12,10)\) self-dual quaternary code. IEEE Trans. Inform. Theory 36, 1153–1156 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  18. MacWilliams F.J., Mallows C.L., Sloane N.J.A.: Generalizations of Gleason’s theorem on weight enumerators of self-dual codes. IEEE Trans. Inform. Theory 18, 794–805 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  19. MacWilliams F.J., Odlyzko A.M., Sloane N.J.A., Ward H.N.: Self-dual codes over GF(\(4\)). J. Comb. Theory Ser. A 25, 288–318 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  20. Mallows C.L., Sloane N.J.A.: An upper bound for self-dual codes. Inform. Control 22, 188–200 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  21. Mallows C.L., Pless V., Sloane N.J.A.: Self-dual codes over \(GF(3)\). SIAM J. Appl. Math. 31, 649–666 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  22. Miezaki T., Munemasa A., Nakasora H.: A note on Assmus-Mattson type theorems. Des. Codes Cryptogr. 89, 843–858 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  23. Nebe G., Villar D.: An analogue of the Pless symmetry codes. In: Seventh International Workshop on Optimal Codes and Related Topics, Bulgaria, pp. 158–163 (2013).

  24. Pless V., Sloane N.J.A., Ward H.N.: Ternary codes of minimum weight \(6\) and the classification of self-dual codes of length \(20\). IEEE Trans. Inform. Theory 26, 305–316 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  25. Rains E., Sloane N.J.A.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998).

    Google Scholar 

  26. Roberts A.M.: Quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40 from modifications of well-known circulant construction. https://arxiv.org/abs/2102.12326. Accessed 28 July 2022.

  27. Russeva R.P.: Self-dual \([24, 12, 8]\) quaternary codes with a nontrivial automorphism of order \(3\). Finite Fields Appl. 8, 34–51 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  28. Seberry J., Yamada M.: Hadamard Matrices: Constructions using Number Theory and Linear Algebra. Wiley, Hoboken, NJ (2020).

    Book  MATH  Google Scholar 

  29. Wolfram Research, Inc.: Mathematica, Version 12.3.1, https://www.wolfram.com/mathematica (2022).

  30. Zhang S.: On the nonexistence of extremal self-dual codes. Discret. Appl. Math. 91, 277–286 (1999).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers 19H01802 and 21K03350.

Author information

Authors and Affiliations

  1. Department of Computer Science, Shizuoka University, Hamamatsu, 432–8011, Japan

    Makoto Araya

  2. Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, 980–8579, Japan

    Masaaki Harada

Authors
  1. Makoto Araya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masaaki Harada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Makoto Araya.

Ethics declarations

Conflict of interest

The authors declare there are no conflicts of interest.

Additional information

Communicated by J. Bierbrauer.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Araya, M., Harada, M. Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes. Des. Codes Cryptogr. 91, 1813–1843 (2023). https://doi.org/10.1007/s10623-022-01172-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-022-01172-7

Keywords

Mathematics Subject Classification

Search

Navigation