We study function computation over a Gaussian multiple-access channel (MAC), where multiple transmitters aim at computing a function of their values at a common receiver. To this end, we propose a novel coded-modulation framework for over-the-air computation (OAC) based on hierarchical constellation design, which supports reliable computation of multiple function outputs using a single channel use. Moreover, we characterize the achievable computation rate and show that the proposed hierarchical constellations can compute R output functions with decoding error probability epsilon while the gap to the optimal computation rate scales as O(\log_2(1/ε)/K) for independent source symbols, where K denotes the number of transmitters. Consequently, this gap vanishes as the network size grows, and the optimal rate is asymptotically attained. Furthermore, we introduce a shielding mechanism based on variable-length block coding that mitigates noise-induced error propagation across constellation levels while preserving the superposition structure of the MAC. We show that the shielding technique improves reliability, yielding a gap that scales optimally as O(\log_2\ln{(1/ε)}), regardless of the source distribution. Together, these results identify the regimes in which uncoded or lightly coded OAC is information-theoretically optimal, providing a unified framework for low-latency, channel-agnostic function computation.

翻译：我们研究了高斯多址信道上的函数计算问题，其中多个发射机旨在通过公共接收机计算其数值的函数。为此，我们提出了一种基于分层星座设计的无线计算新型编码调制框架，该框架支持在单次信道使用中可靠计算多个函数输出。此外，我们刻画了可达计算速率，并证明了对于独立源符号，所提出的分层星座能以解码错误概率ε计算R个输出函数，同时与最优计算速率之间的差距按O(log₂(1/ε)/K)缩放，其中K表示发射机数量。因此，随着网络规模增大，该差距趋于零，从而渐近达到最优速率。进一步，我们引入了一种基于可变长度分组编码的屏蔽机制，该机制在保持多址信道叠加结构的同时，减轻了噪声引起的跨星座层级错误传播。我们证明了屏蔽技术提高了可靠性，产生的差距按最优速率O(log₂ln(1/ε))缩放，且与源分布无关。综上，这些结果确定了未编码或轻编码无线计算在信息论意义下最优的适用场景，为低延迟、信道无关的函数计算提供了一个统一框架。