A generator coordinate version of the Dirac-Fock equations for relativistic closed-shell atoms is presented. The integration of the Dirac-Fock equations is performed through the integral discretization technique so as to preserve the continuous character of the generator coordinate formalism. With the new formalism we generate a universal Gaussian basis set for relativistic closed-shell atoms with d and f orbitals (zinc up to nobelium). The results obtained with the universal Gaussian basis set for Dirac-Fock-Coulomb self-consistent-held energies are compared with numerical-finite-difference results and Dirac-Fock-Coulomb energies obtained by using other Gaussian basis sets. The excellent performance of our universal Gaussian basis set is attributed to the integral discretization technique of the generator coordinate Dirac-Fock method, as with it we are capable of generating Gaussian-type function exponents that are able to represent properly the relativistic kinematics of an electron inside the nucleus.