  We construct a family of n+1 dyadic filtrations in R^n, so that every
Euclidean ball B is contained in some cube Q of our family satisfying diam(Q)
\le c_n diam(B) for some dimensional constant c_n. Our dyadic covering is
optimal on the number of filtrations and improves previous results of Christ
and Garnett/Jones by extending a construction of Mei for the n-torus. Based on
this covering and motivated by applications to matrix-valued functions, we
provide a dyadic nondoubling Calder\'on-Zygmund decomposition which avoids
Besicovitch type coverings in Tolsa's decomposition. We also use a recent
result of Hyt\"onen and Kairema to extend our dyadic nondoubling decomposition
to the more general setting of upper doubling metric spaces.
