  Device independent dimension witnesses (DW) are a remarkable way to test the
dimension of a quantum system in a prepare-and-measure scenario imposing
minimal assumptions on the internal features of the devices. However, as the
dimension increases, the major obstacle in the realization of DW arises due to
the requirement of many outcome quantum measurements. In this article, we
propose a new variant of a widely studied communication task (random access
code) and take its average payoff as the DW. The presented DW applies to
arbitrarily large quantum systems employing only binary outcome measurements.
