Weight enumerators of self-dual quantum codes

We use algebraic invariant theory to study three weight enumerators of formally self-dual quantum codes over any finite fields. We show that the weight enumerator of a formally self-dual quantum code can be expressed algebraically by two polynomials and the double weight enumerator of a formally self-dual quantum code can be expressed algebraically by five polynomials. We also explicitly compute the complete weight enumerators of some special formally self-dual quantum codes. Our approach avoids applying the well-known Molien's formula and demonstrates the potential of employing algebraic invariant theory to compute weight enumerators of quantum codes.