The objective of this paper is the so-called open discharge in the pulsed overvoltage regime generating the high-current electron beam of runaway electrons. Due to the modification of the electrode system of the open discharge by introducing of the third electrode, we were able to appreciably improve the transverse homogeneity of the generated e-beam. Besides, this novelty allows us to increase both the stability of this discharge and energy of runaway electrons up to 25 keV. The extended set of experimental data characterizing the overvoltage regime of the pulsed open discharge in deuterium at low pressure is presented. The mechanisms responsible for the breakdown, sustaining and instability of the strongly overvoltage pulsed discharge in a narrow gap are also discussed on the basis of the performed numerical calculations.