The Atom Conspiracy by Jeff Sutton is a rare blend of mystery and science fiction which usually don’t work together; this is one of the best examples I’ve found, though it’s not exactly a “whodunnit.” It’s a detective / political thriller with science fiction elements important to the plot, set on Earth about 500 years in the future.
After a nuclear war had left survivors in a few areas, primarily Australia, a new world government, with high IQ leaders, outlawed all nuclear research for any purpose. Meanwhile espers (mutant telepaths) evolve and are shunned and outlawed, mostly in hiding (reminding me of A.E. Van Vogt’s SF classic Slan).
In 2449 A.D. someone is found dead from radiation exposure. The government’s leader secretly enlists an average police detective, Max Krull, to investigate the suspected conspiracy.
While Krull’s investigation begins to uncover connections to the espers another government faction frames him as a conspirator, and most of the novel follows his continued investigation while evading them and hiding, with plenty of tension and plot twists. The issue of the three-class society based on IQ is also interwoven with the mystery.
Tension mounts as an election nears and the bad guys close in on Krull, building to an ending full of surprises in which the science fiction elements are integral but don’t spoil the mystery (which is the problem with other science-fiction/mysteries). It’s well written and still works today, just by pushing the dates another 50 years into the future.