halacha - Can the entire succah be constructed by a Gentile?

In many shuls, I see that the hired Gentile custodian constructs the entire succah including laying down the schach.

I understood that some part of the succah must be done by a Jew - at the least, laying down the schach. I also understood that the Shehechiyanu brachah said during the first night Kiddush is partly for the mitzvah of constructing the succah.

Can a Gentile construct the entire succah? Is he considered a shaliach mitzvah? (I gather that a Gentile can never be a shaliach, as he is not obligated in this type of mitzvah, anyway.)