  Starting with a product initial state, squeezed (squeezed coherent) state in
one of the modes, and vacuum in the rest, we report that a circular waveguide
comprising modes coupled with varying coupling strength is capable of producing
genuine multimode entanglement (GME), quantified via the generalized geometric
measure (GGM). We demonstrate that for a fixed coupling and squeezing strength,
the GME content of the resulting state increases as the range of couplings
between the waveguides increases, although the GGM collapses and revives with
the variation of coupling strength and time. The advantage of long-range
coupling can be emphasized by measuring the area under the GGM curve, which
clearly illustrates the growing trends of GME with the increasing range of
couplings. Moreover, long-range couplings help in generating a higher GGM for a
fixed coupling strength. We analytically determine the exact expression of GGM
for systems involving an arbitrary number of modes, when all the modes interact
with each other equally. The entire analysis is performed in the phase-space
formalism. We manifest the constructive effect of disorder in the coupling
parameter, which promises a steady production of GME, independent of the
coupling strength.
