Polynomial invariants of quantum codes

The weight enumerators (Shor and Laflamme 1997) 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_kⁿ. 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))₂ code are uniquely determined, an important step in a proof that any ((4, 4, 2))₂ code is additive (Rains 1999)