We present a comparatively simple way to construct Martin-Löf random and
rec-random sets with certain additional properties, which works by
diagonalizing against appropriate martingales. Reviewing the result
of Gács and Kučera, for any given set X we construct a Martin-Löf
random set from which X can be decoded effectively.
By a variant of the basic construction we obtain a rec-random set
that is weak truth-table autoreducible and we observe that there are
Martin-Löf random sets that are computably enumerable
self-reducible. The two latter results complement the known facts
that no rec-random set is truth-table autoreducible and that no Martin-Löf
random set is Turing-autoreducible [ebert-merkle-vollmer-2002,trakhtenbrot-1970].