In the paper
Eric Rains talks about $ GF(4) $ linear codes and proves some of their properties, for example
"many codes of interest (e.g., GF(4)-linear codes) are built out of distance 2 codes."
"Since GF(4)-linear codes are built out of [[2n, 2(n−1), 2]]s, we find, for instance, that any equivalence between GF(4)-linear codes (subject to certain trivial restrictions) must lie in the Clifford group."
"Corollary 14. Any equivalence of GF(4)-linear quantum codes lies in the Clifford group, unless the codes have minimum distance 1, or contain a codeword of weight 2."
"Lemma 15. A GF(4)-linear code C is spanned by its minimal codewords."
"Corollary 16. If Q is a GF(4)-linear code, then every automorphism of Q lies in the Clifford group."
I'm trying to better understand the limited scope of these results.
What is an example of a stabilizer code which is not $ GF(4) $ linear?
Update: The comment from @unknown says that a CSS code is $ GF(4) $ linear if and only if $ H_x=H_z $. Now I'm curious if the $ [[5,1,3]] $ code is $ GF(4) $ linear.