Cayley graphs of given degree and diameters 3, 4 and 5

Let Cd,k be the largest number of vertices in a Cayley graph of degree d and diameter k. We show that Cd,3≥316(d−3)3 and Cd,5≥25(d−74)5 for any d≥8, and Cd,4≥32(d−85)4 for any d≥10. For sufficiently large d our graphs are the largest known Cayley graphs of degree d and diameters 3, 4 and 5.