  We study the higher-spin gauge theory in six-dimensional anti-de Sitter space
$AdS_6$ that is based on the exceptional Lie superalgebra $F(4)$. The relevant
higher-spin algebra was constructed in arXiv:1409.2185 [hep-th]. We determine
the spectrum of the theory and show that it contains the physical fields of the
Romans $F(4)$ gauged supergravity. The full spectrum consists of an infinite
tower of unitary supermultiplets of $F(4)$ which extend the Romans multiplet to
higher spins plus a single short supermultiplet.
  Motivated by applications to this novel supersymmetric higher-spin theory as
well as to other theories, we extend the known one-loop tests of $AdS/CFT$
duality in various directions. The spectral zeta-function is derived for the
most general case of fermionic and mixed-symmetry fields, which allows one to
test the Type-A and B theories and supersymmetric extensions thereof in any
dimension. We also study higher-spin doubletons and partially-massless fields.
While most of the tests are successfully passed, the Type-B theory in all even
dimensional anti-de Sitter spacetimes presents an interesting puzzle: the free
energy as computed from the bulk is not equal to that of the free fermion on
the CFT side, though there is some systematics to the discrepancy.
