Codes over Z2m and Jacobi Forms of Genus n

The relations between the complete weight enumerators in genus n of Type II codes over Z2m and Jacobi forms of genus n have been discussed. One derives a map between the invariant spaces of the groups G2m, n (or H2m, n, respectively) and the rings of Jacobi forms (or Siegel modular forms, repectively) of genus n. The MacWilliams identities of the complete weight enumerators in genus n of Type II codes over Z2m imply that the complete weight enumerators in genus n of Type II codes over Z2m is in the invariant space of the groups G2m, n.