We propose a simple condition on a distribution of higher codimension
that generalizes this geometric structure.
Geometrically, our condition is satisfied, for example, for the Carnot Caratheodory structure on the boundary
of hyperbolic space, or for a natural codimension 3 distribution on the boundary of domains in hyperkahler manifolds.
We characterize these higher codimension "contact structures" in terms of a property of the appropriate
Heisenberg calculus. We construct a natural generalization of the tangential CR operator, and prove that
it has similar spectral properties as its classical counterpart. There is a Szego projector, and the Boutet de Monvel
index theorem holds for the associated Toeplitz operators.