In subsystem codes, it is well-known that we can extract syndromes by measuring some gauge operators, instead of measuring stabilizers. For example, if a stabilizer $s$ is a product of gauge operators $g_1, g_2,$ and $g_3$, i.e. $s=g_1g_2g_3$, we can obtain the eigenvalue of $s$ by measuring $g_1, g_2,$ and $g_3$ and taking the parities of them. I understand this method.
Although the method above works just for extracting syndrome once, if we measure gauge operators, they become stabilizers. It corresponds to gauge fixing. Thus, in the situations where we need to repeat syndrome extraction, it seems that the code changes before we first extract syndrome and after that. If $s$ is the stabilizer generator before we measure it, stabilizer generators becomes $s, g_1, g_2, g_3$ after we extract the syndrome.
The question is that, is my understanding above true? If so, does the way of implementing logical gates change after we extract syndrome? Also, is it true that in the first round of syndrome extraction, we use only the value of $s$ in decoding, but after that, we use the values of the three of $s, g_1, g_2, g_3$ (because three of them are stabilzer generatos)?
