Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of François Jaeger (a survey)

Bannai, Eiichi

Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 763-782.

La propriété d’invariance modulaire des schémas d’association est rappelée en connexion avec notre travail commun avec François Jaeger. Puis nous considérons les codes sur $F_2$ examinant comment par leurs divers types d’énumérateurs de poids ils sont reliés à divers types de formes modulaires par les invariants polynomiaux de certains groupes finis et les thêta-séries. Récemment on a considéré et étudié intensivement non seulement le cas des codes sur un corps fini arbitraire mais aussi les codes sur des anneaux finis et des groupes finis. Nous montrons comment la détermination des solutions de la propriété d’invariance modulaire des groupes finis abéliens (notre travail commun avec François Jaeger) est utilisée pour définir le concept de code de type II sur un groupe fini abélien arbitraire. Nous donnons comme exemple de l’utilité de ces codes de type II une application aux formes modulaires hermitiennes.

Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over $F_2$ discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied extensively. We show how the determination of the solutions of the modular invariance property of finite abelian groups (our joint work with Jaeger) is used to define the concept of Type II codes over arbitrary finite abelian groups. As an example of the usefulness of such Type II codes, we give an application to hermitian modular forms.

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