# enumerators polynomials and distance for a quantum error correcting code

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: Define weight enumerators \begin{align*} A_i &= \frac{1}{4^k} \sum_{E \in \mathcal{E}_i} |Tr(E \Pi)|^2 \\ B_i &= \frac{1}{2^k} \sum_{E \in \mathcal{E}_i} Tr( E \Pi E \Pi) \end{align*} Here $$\Pi$$ is the code projector and $$\mathcal{E}_i$$ are the Pauli errors with weight $$i$$. Then the code has distance $$d$$ if and only if $$A_i=B_i$$ for all $$i \leq d-1$$. In particular note that $$A_0=1=B_0$$ always. So the code has distance $$2$$ if and only if $$A_1-B_1=0$$ And has distance $$3$$ if and only if $$A_1-B_1=0=A_2-B_2$$ As noted in the answer below, the distance $$d$$ is the degree of the first nonzero term in the polynomial $$B-A$$.

For a classical code the enumerator gives the distance directly $$p_C = 1 + n_d x^d + \cdots$$. So the first nonzero power is the distance; $$n_d$$ is the number of codewords at that disatance. For a qauntum code there are two polynomials at play : $$p_H$$ for the stabilizer code and $$p_N$$ for its normalizer. $$p_N-p_H = n_d x^d + \cdots$$ gives you the distance. (Assuming linear classical codes and stabilizer codes here)
Here's an example for the $$[[5,1,3]]$$ code :
The stabilizer is generated by 4 elements; calculating the weights of the corresponding (16 element) group gives $$p_H = 1 + 15 x^4$$. Note sum of coeffecients is 16.
The normalizer is generated by 6 elements (4 stabilizers + 2 logicals); calculating the weights of the corresponding (64 element group) gives $$p_N=1+ 30 x^3 + 15 x^4 + 18 x^5$$. Note sum of coefficients is 64.
So $$p_N - p_H = 30x^3 +18 x^5$$ and the distance is 3.
• could you say more about the definition of $p_H$ and $p_N$? Dec 11, 2022 at 21:04