  We propose a novel randomization approach for randomized controlled trials
(RCTs), named the cube method. The cube method allows for the selection of
balanced samples across various covariate types, ensuring consistent adherence
to balance tests and, whence, substantial precision gains when estimating
treatment effects. We establish several statistical properties for the
population and sample average treatment effects (PATE and SATE, respectively)
under randomization using the cube method. The relevance of the cube method is
particularly striking when comparing the behavior of prevailing methods
employed for treatment allocation when the number of covariates to balance is
increasing. We formally derive and compare bounds of balancing adjustments
depending on the number of units $n$ and the number of covariates $p$ and show
that our randomization approach outperforms methods proposed in the literature
when $p$ is large and $p/n$ tends to 0. We run simulation studies to illustrate
the substantial gains from the cube method for a large set of covariates.
