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Type II Codes over \(\mathbb{Z}/2k\mathbb{Z}\), Invariant Rings and Theta Series

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Abstract

We consider type II codes over finite rings \(\mathbb{Z}/2k\mathbb{Z} \). It is well-known that their gth complete weight enumerator polynomials are invariant under the action of a certain finite subgroup of \(GL((2k)^g,\mathbb{C})\), which we denote Hk,g. We show that the invariant ring with respect to Hk,g is generated by such polynomials. This is carried out by using some closely related results concerning theta series and Siegel modular forms with respect to \(Sp(g,\mathbb{Z})\).

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References

  1. A. Andrianov and V. G. Zhuravlev, Modular Forms and Hecke Operators, Translations of Mathematical Monographs, Vol. 145, Am. Math. Soc., Providence (1995).

  2. E. Bannai S.T. Dougherty M. Harada M. Oura (1999) ArticleTitleType II codes, even unimodular lattices and invariant rings IEEE Trans. Inform. Theory. 45 1194–1205 Occurrence Handle10.1109/18.761269

    Article  Google Scholar 

  3. E. Freitag, Singular Modular Forms and Theta Relations, Lecture Notes in Math., Vol. 1487, Springer Verlag, Berlin/Heidelberg/New York (1991).

  4. J. Igusa, Theta Functions, Grundlehren Math. Wiss., Vol. 194, Springer Verlag, Berlin/Heidelberg/New York (1972).

  5. D. Mumford, Tata Lectures on Theta, I, Progress in Mathematics, Vol. 28, Bikhauser, Boston/Basel/Stuttgart (1983).

  6. D. Mumford, Tata Lectures on Theta, III, (with the collaboration of M. Nori and P. Newman), Progress in Mathematics, Vol. 97, Bikhauser, Boston/Basel/Stuttgart (1991).

  7. G. Nebe, E. M. Rains and N. J. A. Sloane, Codes ad invariant theory (preprint, available at the web address: http://www.mathematik.uni-ulm.de/ReineM/nebe/papers/cliffannounce.ps).

  8. G. Nebe, H.-G. Quebbemann, E. M. Rains and N. J. A. Sloane, Complete weight enumerators of generalized doubly even self-dual codes (preprint, available at the web address: http://www.mathematik.uni-ulm.de/ReineM/nebe/papers/doublyeven.ps, will appear on Finite Fields and Their Applications).

  9. G. Nebe E.M. Rains N.J.A. Sloane (2001) ArticleTitleThe invariants of the Clifford groups Des. Codes, Cryptogr. 24 IssueID1 99–122

    Google Scholar 

  10. B. Runge (1996) ArticleTitleCodes and Siegel modular forms Discrete Math. 148 175–204 Occurrence Handle10.1016/0012-365X(94)00271-J

    Article  Google Scholar 

  11. B. Runge (1995) ArticleTitleTheta functions and Siegel-Jacobi forms Acta Math. 175 165–196

    Google Scholar 

  12. R. Salvati Manni (1991) ArticleTitleThetanullwerte and stable modular forms II Am. J. Math. 113 733–756

    Google Scholar 

  13. R.P. Stanley (1979) ArticleTitleInvariant of finite groups and their applications to combinatorics Bull. Am. Math. Soc. (N.S.). 1 IssueID3 475–511

    Google Scholar 

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  1. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, D-69120, Heidelberg, Germany

    F. L. Chiera

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  1. F. L. Chiera
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Chiera, F.L. Type II Codes over \(\mathbb{Z}/2k\mathbb{Z}\), Invariant Rings and Theta Series. Des Codes Crypt 36, 147–158 (2005). https://doi.org/10.1007/s10623-004-1701-9

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  • DOI: https://doi.org/10.1007/s10623-004-1701-9

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