Note on the g-fold joint weight enumerators of self-dual codes over Z(k)

Recently there has been interest in self-dual codes over finite rings. In t his note, g-fold joint weight enumerators of codes over the ring Z (k) of i ntegers module k are introduced as a generalization of the biweight enumera tors. We establish the MacWilliams relations for these weight enumerators a nd investigate the biweight enumerators of self-dual codes over Z (k). We d erive Gleason-type theorems for the corresponding biweight enumerators with the help of invariant theory.