  The pairing properties of the neutrinoless double-$\beta$ decay candidate
$^{116}$Cd have been investigated. Measurements of the two-neutron removal
reactions on isotopes of $^{114,116}$Cd have been made in order to identify
0$^+$ strength in the residual nuclei up to $\approx$3 MeV. No significant
$L=0$ strength has been found in excited states indicating that the
Bardeen-Cooper-Schrieffer (BCS) approximation is a reasonable basis to describe
the neutrons in the ground state. This approximation avoids complications in
calculations of double-$\beta$ decay matrix elements that use the quasiparticle
random-phase approximation (QRPA) techniques. However this is not the case for
the protons, where pair vibrations are prevalent and the BCS approximation is
no longer valid, complicating the use of traditional QRPA techniques for this
system as a whole.
