The double-beta decay of $^{116}$Cd has been investigated with the help of radiopure enriched $^{116}$CdWO$_4$ crystal scintillators (mass of 1.162 kg) at the Gran Sasso underground laboratory. The half-life of $^{116}$Cd relatively to the $2 u2beta$ decay to the ground state of $^{116}$Sn was measured with the highest up-to-date accuracy as $T_{1/2}=(2.63^{+0.11}_{-0.12})times10^{19}$ yr. A new improved limit on the 0$ u$2$beta$ decay of $^{116}$Cd to the ground state of $^{116}$Sn was set as $T_{1/2}geq 2.2 times 10^{23}$ yr at 90% C.L., which is the most stringent known restriction for this isotope. It corresponds to the effective Majorana neutrino mass limit in the range $langle m_ uranglele(1.0-1.7)$ eV, depending on the nuclear matrix elements used in the estimations. New improved half-life limits for the 0$ u$2$beta$ decay with majoron(s) emission, Lorentz-violating $2 u2beta$ decay and $2beta$ transitions to excited states of $^{116}$Sn were set at the level of $T_{1/2}geq 10^{20}-10^{22}$ yr. New limits for the hypothetical lepton-number violating parameters (right-handed currents admixtures in weak interaction, the effective majoron-neutrino coupling constants, R-parity violating parameter, Lorentz-violating parameter, heavy neutrino mass) were set.
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