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

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 $P/4$.