A brain-computer interface (BCI) is a system that allows a human operator to use only mental commands in controlling end effectors that interact with the world around them. Such a system consists of a measurement device to record the human user's brain activity, which is then processed into commands that drive a system end effector. BCIs involve either invasive measurements which allow for high-complexity control but are generally infeasible, or noninvasive measurements which offer lower quality signals but are more practical to use. In general, BCI systems have not been developed that efficiently, robustly, and scalably perform high-complexity control while retaining the practicality of noninvasive measurements. Here we leverage recent results from feedback information theory to fill this gap by modeling BCIs as a communications system and deploying a human-implementable interaction algorithm for noninvasive control of a high-complexity robot swarm. We construct a scalable dictionary of robotic behaviors that can be searched simply and efficiently by a BCI user, as we demonstrate through a large-scale user study testing the feasibility of our interaction algorithm, a user test of the full BCI system on (virtual and real) robot swarms, and simulations that verify our results against theoretical models. Our results provide a proof of concept for how a large class of high-complexity effectors (even beyond robotics) can be effectively controlled by a BCI system with low-complexity and noisy inputs.