Skip to main content
Springer Nature Link
Log in

Binary self-dual codes and Jacobi forms over a totally real subfield of \({\mathbb {Q}}(\zeta _8)\)

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Let \(K={\mathbb {Q}}(\zeta _8)\) be the complex multiplication field over \({\mathbb {Q}}\) of extension degree 4. We give an integral lattice construction on \({\mathbb {Q}}(\zeta _8)\) induced from binary codes. We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a binary code. If C is a binary Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield \(k={\mathbb {Q}}(\zeta _8+\zeta _8^{-1})\) of K. Also, we see that Hilbert-Siegel modular form of weight l and genus g can be seen in terms of the complete joint weight enumerator for codes \(C_j\), \(1\le j\le g\) over \({\mathbb {F}}_2\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Ankur. Self-dual codes over the ring GR \((p^m, g)\) and Jacobi forms. Asian Eur. J. Math. 10, 1750055 (2017)

  2. Ankur and Kewat, P. K.: Type I and Type II codes over the ring \({F }_ 2+ u {F }_ 2+ v {F }_2+uv{F }_ 2\). Asian Eur. J. Math. 12(2), 1950025 (2019)

  3. Bachoc, C.: Applications of coding theory to the construction of modular lattices. J. Comb. Theory Ser. A 78(1), 92–119 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bannai, E., Dougherty, S.T., Harada, M., Oura, M.: Type II codes, even unimodular lattices, and invariant rings. IEEE Trans. Inf. Theory 45(4), 1194–1205 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Betsumiya, K., Choie, Y.: Jacobi forms over totally real fields and Type II codes over Galois rings GR\((2^m, f)\). Eur. J. Comb. 25(4), 475–486 (2004)

    Article  MATH  Google Scholar 

  6. Betsumiya, K., Choie, Y.: Codes over \({{\mathbb{F}}}_4\), Jacobi forms and Hilbert-Siegel modular forms over \({\mathbb{Q}}\). Eur. J. Comb. 26(5), 629–650 (2005)

    Article  MATH  Google Scholar 

  7. Choie, Y., Dougherty, S.T.: Codes over rings, complex lattices and Hermitian modular forms. Eur. J. Comb. 26(2), 145–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choie, Y., Jeong, E.: Jacobi forms over totally real fields and codes over \({{\mathbb{F}}}_p\). Ill. J. Math. 46(2), 627–643 (2002)

    MATH  Google Scholar 

  9. Choie, Y., Kim, H.: Codes over \({{\mathbb{Z}}}_{2m}\) and Jacobi forms of genus\(n\). J. Comb. Theory Ser. A 95(2), 335–348 (2001)

    Article  MATH  Google Scholar 

  10. Choie, Y., Kim, N.: The complete weight enumerator of Type II code over \({{\mathbb{Z}}}_{2m}\) and Jacobi forms. IEEE Trans. Inf. Theory 47(1), 5397 (2001)

    Google Scholar 

  11. Ebeling, W.: Lattices and codes. A course partially based on lectures by F. Hirzebruch. In Lattices and Codes. Springer, (2013)

  12. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  13. Gritsenko, V.A.: The action of modular operators on the Fourier-Jacobi coefficients of modular forms. Sbornik Math. 47(1), 237–267 (1984)

    Article  MATH  Google Scholar 

  14. Haverkamp, K.K.: Hermitian Jacobi forms. Results Math. 29(1–2), 78–89 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kositwattanarerk, W., Ong, S.S., Oggier, F.: Wiretap encoding of lattices from number fields using codes over \({{\mathbb{F}}}_p\). In: IEEE international symposium on information theory, pp. 2612–2616 (2013)

  16. MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. Elsevier, Amsterdam (1977)

    MATH  Google Scholar 

  17. Serre, J.P.: A Course in Mathematics, vol. 7. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  18. Skogman, H.: Jacobi forms over totally real number fields. Results Math. 39(1–2), 169–182 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics (School of Advanced Sciences), VIT-AP University, Amaravati, Andhra Pradesh, 522237, India

    Ankur

  2. Department of Mathematics & Computing, Indian Institute of Technology(ISM), Dhanbad, 826 004, India

    P. K. Kewat

Authors
  1. Ankur
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. P. K. Kewat
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ankur.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ankur, Kewat, P.K. Binary self-dual codes and Jacobi forms over a totally real subfield of \({\mathbb {Q}}(\zeta _8)\). AAECC 34, 377–392 (2023). https://doi.org/10.1007/s00200-021-00509-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-021-00509-4

Keywords

Mathematics Subject Classification