Hemispherical Concentration Subset Recovery in Many-Access Gaussian Multiple-Access Channels

Abstract

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 \( K_a(n)=\beta n \) for a constant \( \beta > 0 \), and the codebook size is \( M_n=n^d \) with \( d>2 \). We identify a geometric property showing that, for \( 0<\beta<2 \), 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 \( \beta < 1/4 \). The overlap between the reliable decoding range of \( \beta \) 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 \( P/4 \).