Hemispherical Concentration Subset Recovery in Many-Access Gaussian Multiple-Access Channels
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914525735813120 |
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| author | Mirhosseini, Nazanin |
| author_facet | Mirhosseini, Nazanin |
| contents | We consider subset recovery in the many-access Gaussian multiple-access channel with a shared spherical codebook, where codewords are drawn independently and uniformly from the hypersphere of radius \( \sqrt{nP} \), the number of active users scales linearly with the blocklength $n$ as \( K_a(n)=βn \) for a constant \( β> 0 \), and the codebook size is \( M_n=n^d \) with \( d>2 \). We identify a geometric property showing that, for \( 0<β<2 \), any transmitted \( K_a(n) \)-subset lies in a single hemisphere with high probability for sufficiently large $n$. We further show that reliable decoding is possible only for \( β< 1/4 \). The overlap between the reliable decoding range of \( β\) and the hemispherical concentration range motivates our approach of two-stage decoding procedure. In the pre-filtering stage, the decoder restricts attention to a sequence of spherical caps \( \{ \hat{\mathcal{H}}_n \} \) that converges in Hausdorff distance to the hemisphere $\hat{\mathcal{H}}$, whose axis is the normalized observation \( \hat{\mathbf{u}}=\mathbf{Y}/\|\mathbf{Y}\| \). In the second stage, maximum-likelihood decoding is performed over the reduced candidate set. We show that the per-user error probability of the pre-filtering stage vanishes as \( n\to\infty \). Moreover, the per-user error probability of the maximum-likelihood stage over the reduced search space decays exponentially with asymptotic exponent \( P/4 \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03987 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hemispherical Concentration Subset Recovery in Many-Access Gaussian Multiple-Access Channels Mirhosseini, Nazanin Information Theory We consider subset recovery in the many-access Gaussian multiple-access channel with a shared spherical codebook, where codewords are drawn independently and uniformly from the hypersphere of radius \( \sqrt{nP} \), the number of active users scales linearly with the blocklength $n$ as \( K_a(n)=βn \) for a constant \( β> 0 \), and the codebook size is \( M_n=n^d \) with \( d>2 \). We identify a geometric property showing that, for \( 0<β<2 \), any transmitted \( K_a(n) \)-subset lies in a single hemisphere with high probability for sufficiently large $n$. We further show that reliable decoding is possible only for \( β< 1/4 \). The overlap between the reliable decoding range of \( β\) and the hemispherical concentration range motivates our approach of two-stage decoding procedure. In the pre-filtering stage, the decoder restricts attention to a sequence of spherical caps \( \{ \hat{\mathcal{H}}_n \} \) that converges in Hausdorff distance to the hemisphere $\hat{\mathcal{H}}$, whose axis is the normalized observation \( \hat{\mathbf{u}}=\mathbf{Y}/\|\mathbf{Y}\| \). In the second stage, maximum-likelihood decoding is performed over the reduced candidate set. We show that the per-user error probability of the pre-filtering stage vanishes as \( n\to\infty \). Moreover, the per-user error probability of the maximum-likelihood stage over the reduced search space decays exponentially with asymptotic exponent \( P/4 \). |
| title | Hemispherical Concentration Subset Recovery in Many-Access Gaussian Multiple-Access Channels |
| topic | Information Theory |
| url | https://arxiv.org/abs/2604.03987 |