  Time-Delay Interferometry (TDI) is the data processing technique that cancels
the large laser phase fluctuations affecting the one-way Doppler measurements
made by unequal-arm space-based gravitational wave interferometers. In a
previous publication we derived TDI combinations that exactly cancel the laser
phase fluctuations up to first order in the inter-spacecraft velocities. This
was done by interfering two digitally-synthesized optical beams propagating a
number of times clock- and counter-clock-wise around the array. Here we extend
that approach by showing that the number of loops made by each beam before
interfering corresponds to a specific higher-order TDI space. In it the
cancellation of laser noise terms that depend on the acceleration and
higher-order time derivatives of the inter-spacecraft light-travel-times is
achieved exactly. Similarly to what we proved for the second-generation TDI
space, elements of a specific higher-order TDI space can be obtained by first
``lifting'' the basis ($\a, \b, \g, X$) of the $1^{\rm st}$-generation TDI
space to the higher-order space of interest and then taking linear combinations
of them with coefficients that are polynomials of the six delays operators.
Higher-Order TDI might be required by future interplanetary gravitational wave
missions whose inter-spacecraft distances vary appreciably with time, in
particular, relative velocities are much larger than those of currently planned
arrays.
