Weight enumerators of self-dual quantum codes

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