Building codes with transversal CCZ using projective geometry and SAT solvers

We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical T/T gates and whose distance is at least 3. These codes start with 8-divisible X-stabilizers defined by projective geometry, which guarantees d Z at least 3. A SAT solver then finds three X-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from n=48 to n=496, with d Z=3 or 4, and illustrate the method with the [[48,3,3]] code Q 48. Additionally, we prove that no CSS code with three logical qubits, Z-distance at least 3, and a quasi-transversal CCZ gate (transversal T with a diagonal Clifford correction) exists below n=39. The [[47,3,3]] code of Jacinto et al. serves as an upper bound, leaving 39 at most n at most46 open.