An $n\overset{p}{\mapsto}m$ random access code (RAC) is an encoding of $n$ bits into $m$ bits such that any initial bit can be recovered with probability at least $p$, while in a quantum RAC (QRAC), the $n$ bits are encoded into $m$ qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function $f$ on any subset of fixed size of the initial bits, which we call $f$-random access codes. We study and give protocols for $f$-random access codes with classical ($f$-RAC) and quantum ($f$-QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement ($f$-EARAC) and Popescu-Rohrlich boxes ($f$-PRRAC). The success probability of our protocols is characterized by the \emph{noise stability} of the Boolean function $f$. Moreover, we give an \emph{upper bound} on the success probability of any $f$-QRAC with shared randomness that matches its success probability up to a multiplicative constant (and $f$-RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.

翻译：$\ overset{ punpsto} mom 随机存取代码 (RAC) 是将 $ 位元编码为 $ 位元的编码, 或将 $ 位元编码为 美元 位元的编码为 美元 元元元的编码为 美元 。 在 量子 RAC (QRAC) 中, 美元 位元编码编码为 美元 。 自其提议以来, RAC 的概念以多种不同的方式被广泛采用, 例如, 允许使用 共享的 纠缠( 所谓的 缓冲 随机调用代码 ), 或恢复 多位位位元 。 在本文件中, 我们将 RAC 的理念推广到 给给给给给给给定的 Boolean 函数值 $ 美元 的值值值 。 我们研究并给出了 $- 兰次访问代码的规程, 与许多不同的资源一起, 例如 私人或共享的 QRC} Qralal_ 时间, 成功率 只能通过 格式 实现 。