The evolution of the magnetic ordering temperature of the 4d 3 perovskites RTcO 3 (R = Ca, Sr, Ba) and its relation with its electronic and structural properties has been studied by means of hybrid density functional theory and Monte Carlo simulations. When compared to the most widely studied 3d perovskites the large spatial extent of the 4d shells and their relatively strong hybridization with oxygen weaken the tendency to form Jahn-Teller like orbital ordering. This strengthens the superexchange interaction. The resulting insulating G-type antiferromagnetic ground state is characterized by large superexchange coupling constants (26-35 meV) and Neél temperatures (750-1200 K). These monotonically increase as a function of the R ionic radius due to the progressive enhancement of the volume and the associated decrease of the cooperative rotation of the TcO 6 octahedra. Intermediately located between manganese and rhenium, technetium (5s 2 4d 5 ) shares with its isovalent neighbors the intriguing possibility to form oxides with a complex structural, electronic, and magnetic phase diagram. However, the rare occurrence of natural Tc (Tc is essentially an artificial product of fission reactions) and the related radioactive risks (Tc is the lightest radioactive element, whose most abundant isotope, 99 Tc, decays with a half-life of 10 5 years), have made investigations of Tc-based oxides very sparse.1-6 Overcoming these difficulties, Avdeev et al. 5 and Rodriguez et al. 6 have recently reported the successful synthesis of fabricated Tc-based perovskites, namely CaTcO 3 and SrTcO 3 . Furthermore, they have shown that these compounds display the anomalously high Neél temperatures (T N ) of 800 K for CaTcO 3 and 1000 K for SrTcO 3 , by far the highest among materials not incorporating 3d transition metals. These results are surprising and challenge our understanding of the magnetic interaction in perovskites. In particular, they pose three fundamental questions: (i) Is the origin of such a large magnetic ordering temperature related to the strong Tc 4d-O p hybridization?, (ii) What is the role played by the structural degrees of freedom? and (iii) Is the enhancement of T N on Ca→Sr substitution related to the observed increase in unit cell volume and the corresponding modification of the internal atomic positions 7,8 ? In the present letter, by using a combination of hybrid density functional theory 9 and Monte Carlo (MC) simulations, 10 we address these issues at the microscopic level through a systematic study of the series RTcO 3 (R = Ca, Sr, Ba). Our aim here is twofold. First, we wish to interpret and understand the experimental findings for CaTcO 3 and SrTcO 3 , which can be only captured by beyond-local density functional theory (DFT), due to the incorrect treatment of the residual exchangecorrelation effects still present in 4d compounds. 11 Second, we anticipate the experiments predicting the properties of the technetiates series end member BaTcO 3 .We have employed the Heyd-Scuseria-Ernzerhof (HSE) 12 scheme ...