Yes, it does. If Khan could look up the code, then Spock's fear that he might have changed it suggests that it is possible to change it in the field, on the fly, with only the resources normally aboard a starship. Spock wouldn't have suggested the possibility otherwise.
Only the resources normally aboard a starship? It seems to me that there are a lot
of resources normally available aboard a starship. In any case, that doesn't mean that it's easy to do. Khan possessed advanced intelligence that almost certainly enabled him to do things that most people couldn't even dream of.