P.S.
Haha. Looks like I missed the elephant: под-строить is really rare (and a little bit strange) in the meaning of "to build barn". It's first and common meaning is "to build up ground/base for some event". For example "to set up trap". This is there logic of "под-" as "above/bottom" works fine "to prepare ground for...". I bet this meaning was meant by test.