Skip to main content
SpringerLink
Log in

Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields

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

Abstract

In this paper we continue the study of codes over imaginary quadratic fields and their weight enumerators and theta functions. We present new examples of non-equivalent codes over rings of characteristic \(p=2\) and \(p=5\) which have the same theta functions. We also look at a generalization of codes over imaginary quadratic fields, providing examples of non-equivalent pairs with the same theta function for \(p=3\) and \(p=5\).

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

Notes

  1. For our context here, by “non-equivalent” codes, we mean codes having different symmetric weight enumerator polynomials.

References

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

    Google Scholar 

  2. Chua K.S.: Codes over \(\rm {GF}(4)\) and F \(_2\times \) F \(_2\) and Hermitian lattices over imaginary quadratic fields. Proc. Am. Math. Soc. 133(3), 661–670 (2005).

  3. Dougherty S., Kim J.-L., Lee Y.: Linear codes, hermitian lattices, and finite rings (in press).

  4. Günther A., Nebe G.: Clifford–Weil groups for finite group rings, some examples. Alban. J. Math. 2(3), 185–198 (2008).

    Google Scholar 

  5. Günther A., Nebe G., Rains E.M.: Clifford–Weil groups of quotient representations. Alban. J. Math. 2(3), 159–169 (2008).

    Google Scholar 

  6. Leech J., Sloane N.J.A.: Sphere packing and error-correcting codes. Can. J. Math. 23, 718–745 (1971).

    Google Scholar 

  7. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. II. North-Holland Mathematical Library, vol. 16, pp. 1–10, 370–762. North-Holland, Amsterdam (1977).

  8. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. I. North-Holland Mathematical Library, vol. 16, pp. 1–15, 1–369. North-Holland, Amsterdam (1977).

  9. Shaska T., Shor C.: Codes over \(F_{p^2}\) and \(F_p \times F_p\), lattices, and corresponding theta functions. Adv. Coding Theory Cryptol. 3, 70–80 (2007).

  10. Shaska T., Wijesiri S.: Codes over rings of size four, Hermitian lattices, and corresponding theta functions. Proc. Am. Math. Soc. 136, 849–960 (2008).

    Google Scholar 

  11. Shaska T., Shor C., Wijesiri S.: Codes over rings of size \(p^2\) and lattices over imaginary quadratic fields. Finite Fields Appl. 16(2), 75–87 (2010).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Oakland University, Rochester, MI , 48309, USA

    T. Shaska

  2. Department of Mathematics, Western New England University, Springfield, MA , 01119, USA

    C. Shor

Authors
  1. T. Shaska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Shor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Shaska.

Additional information

Communicated by A. Winterhof.

Appendix: Full details for given examples with \(p=2\)

Appendix: Full details for given examples with \(p=2\)

In this appendix, we give the specific data for the examples of pairs of non-equivalent codes with the same theta function for \(p=2\). Tables 9, 10, 11, 12 and 13 contain the codes given in Examples 3–7.

Table 9 Two nonequivalent codes of length \(n=3\) over \(\mathbb {F}_2\times \mathbb {F}_2\) with the same theta series for \(\ell =7\) and different theta series for \(\ell >7\)
Full size table
Table 10 Two nonequivalent codes of length \(n=2\) over \(\mathbb {F}_2\times \mathbb {F}_2\) with the same theta series for \(\ell =7\) and different theta series for \(\ell >7\)
Full size table
Table 11 Two nonequivalent codes of length \(n=3\) over \(\mathbb {F}_2\times \mathbb {F}_2\) with the same theta series for \(\ell =7\) and different theta series for \(\ell >7\)
Full size table
Table 12 Two nonequivalent codes of length \(n=3\) over \(\mathbb {F}_2\times \mathbb {F}_2\) with the same theta series for \(\ell =7\) and different theta series for \(\ell >7\)
Full size table
Table 13 Two nonequivalent codes of length \(n=4\) over \(\mathbb {F}_4\) with the same theta series for \(\ell =3\) and different theta series for \(\ell >3\)
Full size table

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shaska, T., Shor, C. Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields. Des. Codes Cryptogr. 76, 217–235 (2015). https://doi.org/10.1007/s10623-014-9943-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-014-9943-7

Keywords

Mathematics Subject Classification

Search

Navigation