Computing a minimum-size circuit that implements a certain function is a standard optimization task. We consider circuits of CNOT gates, which are fundamental binary gates in reversible and quantum computing. Algebraically, CNOT circuits on n qubits correspond to \(\textsf{GL}(n,2)\) , the general linear group over the field of two elements, and circuit minimization reduces to computing distances in the Cayley graph \(G_n\) of \(\textsf{GL}(n,2)\) generated by transvections. However, the super-exponential size of \(\textsf{GL}(n,2)\) has made its exploration computationally challenging. In this paper, we develop a new approach for computing distances in \(G_n\) , allowing us to synthesize minimum circuits that were previously beyond reach (e.g., we can synthesize optimally all circuits over \(n=7\) qubits). Towards this, we establish two theoretical results that may be of independent interest. First, we give a complete characterization of all isometries in \(G_n\) in terms of (i) permuting qubits and (ii) swapping the arguments of all CNOT gates. Second, for any fixed d, we establish polynomials in n of degree 2d that characterize the size of spheres in \(G_n\)  at distance d from the identity, as long as \(n\ge 2d\) . With these tools, we revisit an open question of [Bataille, 2020] regarding the smallest number \(n_0\) for which the diameter of \(G_{n_0}\) exceeds \(3(n_0-1)\) . It was previously shown that \(6\le n_0 \le 30\) , a gap that we tighten considerably to \(8\le n_0 \le 20\) . We also confirm a conjecture that long cycle permutations lie at distance \(3(n-1)\) , for all \(n\le 8\) , extending the previous bound of \(n\le 5\) .