The constructions of 1-out-of-$n$ OT for $l$-bit strings I've seen had communication complexity proportional to $nl$. I wonder, is it possible to do OT with active security and transfer less than $O(nl)$ bits (I'm ignoring the security parameter in $O$-notation here)? The important part for me here is making it cheaper than just transfering sender's input to the receiver.
Is there some inherent limitation that doesn't allow this kind of OT to transfer less bits than the sender's input? Are there any communication lower bounds for this?