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The Extreme Light Infrastructure - Nuclear Physics (ELI-NP) is under construction in Magurele, Bucharest, Romania. The facility will use two 10 PW lasers and a high intensity, narrow bandwidth gamma beam for stand-alone and combined laser-gamma experiments. The accurate estimation of particle doses and their restriction within the limits for both personel and general public is very important in the design phase of any nuclear facility. In the present work, Monte Carlo simulations are performed using FLUKA and MCNPX to design 19.4 and 4 MeV gamma beam dumps along with shielding of experimental areas. Dose rate contour plots from both FLUKA and MCNPX along with numerical values of doses in experimental area E8 of the facility are performed. The calculated doses are within the permissible limits. Furthermore, a reasonable agreement between both codes enhances our confidence in using one or both of them for future calculations in beam dump designs, radiation shielding, radioactive inventory, and other calculations releated to radiation protection. Residual dose rates and residual activity calculations are also performed for high-energy beam dump and their effect is negligible in comparison to contributions from prompt radiation. [ABSTRACT FROM AUTHOR]