  Random access codes (RACs) are used by a party to despite limited
communication access an arbitrary subset of information held by another party.
Quantum resources are known to enable RACs that break classical limitations.
Here, we study quantum and classical RACs with high-level communication. We
derive average performances of classical RACs and present families of
high-level quantum RACs. Our results show that high-level quantum systems can
significantly increase the advantage of quantum RACs over the classical
counterparts. We demonstrate our findings in an experimental realization of a
quantum RAC with four-level communication.
