  The theory of gravity with an auxiliary extra dimension is known to give the
ghost-free cubic completion of the Fierz-Pauli mass term in the decoupling
limit. Our work generalizes the boundary condition in the auxiliary dimension
that avoids ghosts order-by-order, and to all orders, in the decoupling limit.
Furthermore, we extend the formalism to the case of many auxiliary dimensions,
and we show that the multi-dimensional extension with the rotationally
invariant boundaries of the bulk, is equivalent to the model with a single
auxiliary dimension. The above constructions require the appropriate adjustment
of the boundary condition, which we discuss in detail. The other possible
extension of the original model by the Gauss-Bonnet term is studied as well.
