Random Access Codes: Explicit Constructions, Optimality, and Classical-Quantum Gaps

A random access code (RAC) encodes an $L$-bit string into a $k$-bit message, where $L>k$, such that any requested bit can be decoded with high probability; a quantum RAC (QRAC) replaces the message with $k$ qubits. This paper provides a geometric characterization of optimal classical $(L,k)$-RACs under both average and worst-case success criteria. We show that the average problem reduces to selecting $2^k$ representatives in $\{0,1\}^L$, whereas the worst-case problem reduces to selecting $2^k$ points in $[0,1]^L$ that minimize a distance-like objective. This framework establishes optimality for several parameter families $(L,k)$, with optimal constructions in many cases realized by standard infinite families of binary linear codes. For the parameter family $(2^k-1,k)$, we prove the worst-case optimality of a classical construction and present an explicit QRAC whose worst-case success probability is strictly higher than the classical optimum, thereby establishing a classical--quantum separation for this family. For the parameter family $(L,L-1)$, the framework identifies a classical RAC construction that is optimal under the average criterion and, assuming a stated conjecture, also optimal under the worst-case criterion. As a by-product, the same geometric viewpoint recovers explicit $(L,L-1)$-QRACs similar to existing constructions that attain the value of an upper bound conjectured in prior work to be tight.