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.)