  We investigate the task of $d$-level random access codes ($d$-RACs) and
consider the possibility of encoding classical strings of $d$-level symbols
(dits) into a quantum system of dimension $d'$ strictly less than $d$. We show
that the average success probability of recovering one (randomly chosen) dit
from the encoded string can be larger than that obtained in the best classical
protocol for the task. Our result is intriguing as we know from Holevo's
theorem (and more recently from Frenkel-Weiner's result [Commun. Math. Phys.
340, 563 (2015)]) that there exist communication scenarios wherein quantum
resources prove to be of no advantage over classical resources. A
distinguishing feature of our protocol is that it establishes a stronger
quantum advantage in contrast to the existing quantum $d$-RACs where $d$-level
quantum systems are shown to be advantageous over their classical $d$-level
counterparts.
