I have a system of quadratic forms (homogeneous polynomials of degree $ 2 $) with integer coefficients. Each quadratic form is the trace of a product of matrices. I'm solving for the matrix entries. The quadratic forms are indexed by $ i $ where
$$ Q_i:= K\sum_{E \in \mathcal{E}_i} \operatorname{Tr} \left( E X E^{−1} X \right)- \sum_{E \in \mathcal{E}_i} | \operatorname{Tr} (EX) |^2 $$
where each $ \mathcal{E}_i $ is a different set of orthogonal matrices with integer entries and the variables I am solving for are the entries of the symmetric matrix $ X $. $ K $ is a fixed integer, usually $ 2 $, and is not important.
I am interested in finding any solution to this system of quadratic forms (other than the all $ 0 $ solution). Ideally I would like a solution over the integers but any real solution will do.
Since I need a solution that is exact I'm planning to use Groebner basis. How many variables can I use before the Groebner basis problem becomes slow and hard? If I have $ 100 $ variables is that already going to be impossibly slow?
Are there any special precautions or recommended methods for doing Groebner basis stuff over the integers? Is there any advantage that can be gained from the fact that my polynomials are all quadratic forms?
Motivation
These quadratic forms are the distance conditions for a quantum error correcting code. Basically distance is the number of tensor factors that an error can occur on and still be correctable by the code. In particular, it is well know that more standard descriptions of the distance conditions are equivalent to certain coefficients of the weight enumerator and dual weight enumerator of the code being equal. Explicitly: A code has distance $ d $ if and only if $$ A_i=B_i $$ for all $ 0\leq i \leq d-1 $. Where $$ A_i:=\frac{1}{K^2}\sum_{E \in \mathcal{E}_i} |Tr(EX)|^2 $$ and $$ B_i:=\frac{1}{K}\sum_{E \in \mathcal{E}_i} Tr(EXE^{-1}X) $$ $ \mathcal{E}_i $ are the $ n $ qubit real pauli errors of weight $ i $ (they are signed permutation matrices), $ K $ is the dimensions of the code, and $ X $ is the code space projector. $ n,K $ are integers fixed ahead of time. We are solving for the code space projector $ X $. So the code has distance $ d $ if and only if all the quadratic forms $$ Q_i=B_i-A_i $$ all vanish. For reference see Corollary 9 of Eric M. Rains's paper.