  The weight enumerators (quant-ph/9610040) of a quantum code are quite
powerful tools for exploring its structure. As the weight enumerators are
quadratic invariants of the code, this suggests the consideration of
higher-degree polynomial invariants. We show that the space of degree k
invariants of a code of length n is spanned by a set of basic invariants in
one-to-one correspondence with S_k^n. We then present a number of equations and
inequalities in these invariants; in particular, we give a higher-order
generalization of the shadow enumerator of a code, and prove that its
coefficients are nonnegative. We also prove that the quartic invariants of a
((4,4,2)) are uniquely determined, an important step in a proof that any
((4,4,2)) is additive ([2]).
