  We consider the Sherali-Adams (SA) refutation system together with the
unusual binary encoding of certain combinatorial principles. For the unary
encoding of the Pigeonhole Principle and the Least Number Principle, it is
known that linear rank is required for refutations in SA, although both admit
refutations of polynomial size. We prove that the binary encoding of the
Pigeonhole Principle requires exponentially-sized SA refutations, whereas the
binary encoding of the Least Number Principle admits logarithmic rank,
polynomially-sized SA refutations. We continue by considering a refutation
system between SA and Lasserre (Sum-of-Squares). In this system, the Least
Number Principle requires linear rank while the Pigeonhole Principle becomes
constant rank.
