Updates on alternative pre-booster ring design and wiggler magnet considerations of SPS for the FCC $e^{+}e^{-}$ injector

The Future Circular $e^{+}e^{−}$ Collider (FCC-$e^{+}e^{−}$) injector complex needs to produce and to transport a high-intensity $e^{+}e^{−}$ beam at a fast repetition rate for topping up the collider at its collision energy. Two different options are under consideration as pre-accelerator before the bunches are transferred to the high-energy booster: using the existing SPS and designing a completely new ring. The purpose of this paper is to explore the needs and parameters of the existing SPS, to investigate wiggler magnet options for SPS, and provide an updated study of alternative accelerator ring design with injection and extraction energies of 6 and 20 GeV, respectively. In this study, the parameters of both choices are established, including the optics design, layout update and considerations for non-linear dynamics optimization.