Jacobi Polynomials, Type II Codes, and Designs

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices They are useful for coset weight enumeration and weight enumeration of children We determine them in most interesting cases in length at most and in some cases in length We use them to construct group divisible designs packing designs covering designs and t r designs in the sense of Calderbank Delsarte A major tool is invariant theory of nite groups in particular simultaneous invariants in the sense of Schur polarization and bivariate Molien series A combinatorial interpretation of the Aronhold polarization operator is given New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here