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.

翻译：随机接入码（RAC）将一个$L$比特字符串编码成$k$比特消息（其中$L>k$），使得任意请求比特能以高概率解码；量子随机接入码（QRAC）则将消息替换为$k$量子比特。本文给出了在平均和最坏情况成功准则下最优经典$(L,k)$-RAC的几何刻画。我们证明平均问题可归结为在$\{0,1\}^L$中选择$2^k$个代表点，而最坏情况问题则转化为在$[0,1]^L$中选择$2^k$个点以最小化类距离目标函数。该框架确立了多个参数族$(L,k)$的最优性，其中许多情况下的最优构造由标准二进制线性码的无穷族实现。对于参数族$(2^k-1,k)$，我们证明了经典构造的最坏情况最优性，并给出一个显式QRAC，其最坏情况成功概率严格高于经典最优值，从而在该参数族中建立了经典-量子分离。对于参数族$(L,L-1)$，该框架识别出一个在平均准则下最优的经典RAC构造，并在假设某猜想成立时，该构造在最坏准则下也最优。作为副产品，相同的几何视角恢复了类似于现有构造的显式$(L,L-1)$-QRAC，这些构造达到先前工作中猜想为紧的上界值。