ASBI Pocket Beit Midrash: Laws of Synagogue 150:8

It is taught in the Tosefta in the third chapter of Megillah: We do not make doors into the synagogue other than on the eastern side of the synagogue for this was how it was in the Temple which opened to the east" The reason for this is that they faced west to pray and the entrance was on the east so that when they entered they could bow and face the Aron which was on the western side. And since we are located to the west of Eretz Yisrael, (as I wrote earlier in section 94) we need to have our entrances on the west. And this is what the rishonim all wrote and this is the simple common practice and it should not be changed.

(And the Beit Yosef who writes in the name of the Hagahot Maimoniot: And so too with regards to north and south. See there. And it seems to me that his intention is the same that those who pray facing south should enter the synagogue from the north and vice versa. And from the Levush and the Perishah it seems that none of this is disqualifying. And see there; and this requires further analysis).