  A set of orthogonal multipartite quantum states are called
(distinguishability-based) genuinely nonlocal if they are locally
indistinguishable across any bipartition of the subsystems. In this work, we
consider the problem of constructing small genuinely nonlocal sets consisting
of generalized GHZ states in multipartite systems. For system (C^2)^(\otimes N)
where N is large, using the language of group theory, we show that a tiny
proportion {\Theta}[1/2^(N/2)] of the states among the N-qubit GHZ basis
suffice to exhibit genuine nonlocality. Similar arguments also hold for the
canonical generalized GHZ bases in systems (C^d)^(\otimes N), wherever d is
even and N is large. What is more, moving to the condition that any fixed N is
given, we show that d + 1 genuinely nonlocal generalized GHZ states exist in
(C^d)^(\otimes N), provided the local dimension d is sufficiently large. As an
additional merit, within and beyond an asymptotic sense, the latter result also
indicates some evident limitations of the "trivial othogonality-preserving
local measurements" (TOPLM) technique that has been utilized frequently for
detecting genuine nonlocality.
