that would be CLIC (Compact LInear Collider) or the ILC (International Linear Collider).
Both projects are designing similar machines as LHC, but they are colliding electron - positrons instead of protons. The have to be build straight (as oppose to the circular LHC), since light particles as electrons would lose too much energy if forced to a circular orbit at the high energies CLIC and ILC aims for (TeV and above).
The purpose of both CLIC and ILC is basically to do detailed studes of what LHC will find. LHC can reach higher energies in the collisions (due to heavier particles and circular design), but since protons are made out of "lots of smaller stuff" it is much harder to deduct what actually happend after each collisions
(you can't tell really which "smaller stuff part" of the proton collided with which "smaller stuff part" of the colliding proton, and you don't know how large part of the total energy of the proton belongs to which part....).
Electrons does not, as far as we know, contain smaller building blocks and the result of an collision is much cleaner. (you know exactely which part collided with which (always electron-positron), and you know the energy (simply the combined energy of the electron - positron pair)...
(The tunnel now used for LHC earlier housed an electron collider, LEP)
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