Let me know if this question is too broad:
I'm learning more about enumerator polynomials for quantum error correcting codes (QECCs). What is the relationship between enumerator polynomials and the distance $ d $ of an $ [[n,k,d]] $ quantum code?
For example is there any condition in terms of enumerator polynomials that is sufficient to imply that $ d \geq 2 $ (error detecting) or $ d \geq 3 $ (error correcting)?
Update: here is my understanding of the method described by @unknown :
To any subgroup $ G $ of the the group $ P_m $ on $ m $ qubit Pauli operators we can assign a polynomial called the weight enumerator which has the form $$ p_G:=\sum_{i=1}^m n_i x^{n_i} $$ where $ n_i $ is the number of weight $ i $ Pauli operators (mod global phase, this restriction is important because, for example, it means the degree 0 coefficient is always 1) in the subgroup $ G $.
Let $ V $ be a code space encoded into $ m $ physical qubits. Then let $ N $ be the subgroup of $ P_m $ that maps $ V $ back into $ V $. Let $ S $ be the subgroup of $ P_m $ that stabilizes every element in $ V $. Then by definition the distance of the code is the minimum weight of an element in $ N\S $. Equivalently that is the minimum degree of the polynomial $$ p_N-p_S $$ note that $ S $ is always the Pauli stabilizer of the code while $ N $ is in general not the normalizer of $ S $ if the code is not a stabilizer code. But this setup seems to work in general, even if the code is not a stabilizer code.