Function Computation Over Multiple Access Channels via Hierarchical Constellations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911382667001856 |
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| author | Razavikia, Saeed Kazemi, Mohammad Gündüz, Deniz Fischione, Carlo |
| author_facet | Razavikia, Saeed Kazemi, Mohammad Gündüz, Deniz Fischione, Carlo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12050 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Function Computation Over Multiple Access Channels via Hierarchical Constellations Razavikia, Saeed Kazemi, Mohammad Gündüz, Deniz Fischione, Carlo Information Theory 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. |
| title | Function Computation Over Multiple Access Channels via Hierarchical Constellations |
| topic | Information Theory |
| url | https://arxiv.org/abs/2601.12050 |