The finite Fourier-transform and theta functions

  • Hans Opolka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 229)


The finite Fourier-transform is considered as a linear transformation on a certain space of theta functions and thereby is seen to induce an invertible morphism of Abelian varieties. This is explained in the context of the representation theory of the finite symplectic group. Finally the MacWilliams identities in coding theory are discussed in the light of the theory of theta functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    AUSLANDER, L., TOLIMIERI, R.: Is computing with the finite Fourier-transform pure or applied mathematics?, Bulletin AMS, 1, 1979, 847–897Google Scholar
  2. [2]
    GELBART, S.: Examples of dual reductive pairs, Proc. of Symposia in Pure Math., Vol. 33, part I, AMS, 1979, 287–296Google Scholar
  3. [3]
    GLEASON, A.: Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrès Internat. Math., Vol. 3, Gauthier-Villars, Paris, 1971, 211–215Google Scholar
  4. [4]
    HOWE, R.: On the role of the Heisenberg group in harmonic analysis, Bulletin AMS, 3, 1980, 821–843Google Scholar
  5. [5]
    MUMFORD, D.: Abelian Varieties, Oxford University Press, Oxford, 1970Google Scholar
  6. [6]
    MUMFORD, D.: Tata lectures on theta I, Birkhäuser Verlag, PM 28, Boston, 1983Google Scholar
  7. [7]
    SLOANE, N.J.A.: Binary codes, lattices and sphere packings, in: Combinatorical Surveys: Proc. 6th British Comb. Conference, ed. by P.J. Cameron, Ac. Press, London, 1977, 117–164Google Scholar
  8. [8]
    SLOANE, N.J.A.: Self dual codes and lattices, Proc. of Symposia in Pure Math., Vol. 34, AMS, 1979, 273–308Google Scholar
  9. [9]
    TOLIMIERI, R.: The algebra of the finite Fourier transform and coding theory, Transactions of the AMS, 287, 1985, 253–273Google Scholar
  10. [10]
    WEIL, A.: Sur certains groupes d'opérateurs unitaires, Acta Math., 111, 1964, 143–211Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Hans Opolka
    • 1
  1. 1.Mathematical InstituteUniversity of GöttingenGermany

Personalised recommendations