Double beta decay of $^{116}$Cd has been investigated with the help of radiopure enriched $^{116}$CdWO$_4$ crystal scintillators in the experiment Aurora. The half-life of $^{116}$Cd relatively to the 2$ u$2$beta$ decay of $^{116}$Cd to the ground level of $^{116}$Sn is measured with the highest up-to-date accuracy as $T_{1/2}$ = [2.69 $pm$ 0.02 (stat.) $pm$ 0.14 (syst.)] $times$ 10$^{19}$ yr. A new improved limit on the 0$ u$2$beta$ decay of $^{116}$Cd to the ground state of $^{116}$Sn is set as $T_{1/2}geq 2.4 times 10^{23}$ yr at 90% C.L., that corresponds to the effective Majorana neutrino mass limit in the range $langle$$m_ u$$rangle$ $le$ $(1.1-1.6)$ eV, depending on the nuclear matrix elements used in the estimations. New improved limits on other $2beta$ processes in $^{116}$Cd (decays with majoron emission, transitions to excited levels of $^{116}$Sn) were set at the level of $T_{1/2}geq 10^{21}-10^{22}$ yr.
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