  This paper presents the existence of global solutions to the discrete
Safronov-Dubvoski\v{i} coagulation equations for a large class of coagulation
kernels satisfying $\Lambda_{i,j} = \theta_i \theta_j + \kappa_{i,j}$ with
$\kappa_{i,j} \leq A\theta_i \theta_j, \ \ \forall \ \ i,j\ge 1$ where the
sequence $(\theta_i)_{i\geq 1}$ grows linearly or superlinearly with respect to
$i$. Moreover, the failure of mass-conservation of the solution is also
addressed which confirms the occurrence of the gelation phenomenon.
