Consider an $ [[n,k,d]] $ code. Take the image of the codespace under some Clifford gate $ C $. Does the new code have the same distance?
If $ C $ is a tensor product of single qubit Cliffords, such as $ H $ and $ P $, the answer is yes. Indeed any tensor product of single qubit unitaries preserves distance.
However what about an entangling gate like $ CNOT $? Does the image of the codespace under a $ CNOT $ always have the same distance?
Reflecting on the answer from squiggles a nice counterexample is the following: Consider the $ [[4,2,2]] $ code with stabilizer generators $ XXXX,ZZZZ $. Applying the Clifford gate $ CNOT $ to this code clearly does not preserve distance since the image under $ CNOT $ gives a $ [[4,2,1]] $ stabilizer code with stabilizer generators $ XIXX,IZZZ $.