The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.
"A C.E. Real That Cannot Be SW-Computed by Any Ω Number." Notre Dame J. Formal Logic 47 (2) 197 - 209, 2006. https://doi.org/10.1305/ndjfl/1153858646