Non-asymptotic Sphere-Packing Bounds for Finite-Blocklength Scalar Gaussian Multiple-Access and Broadcast Channels

This paper investigates non-asymptotic achievable rate regions of Gaussian multiple-access channels (MACs) and broadcast channels (BCs) in the finite-blocklength regime. By employing a proposed volume-uniform random coding (VURC) codebook that distributes codewords uniformly within an n-dimensional power sphere, together with successive interference cancellation (SIC) decoding, we derive a closed-form achievability bound, a scheme-specific lower bound on the average error probability, and a fourth-order asymptotic characterization. The proposed SIC achievability bound admits a simple and analytically interpretable closed-form expression, enabling direct performance evaluation without iterative computation. The structure of the proposed SIC asymptotic bound offers a highly extensible framework, allowing advances in single-user finite-blocklength theory to be systematically incorporated into the multiuser setting. Futhermore, to characterize the performance benefit of joint decoding, we further derive numerical and closed-form achievability bounds for the two-user Gaussian MAC under joint nearest-neighbor (JNN) decoding. Numerical results illustrate the performance differences among the proposed SIC and JNN bounds and existing benchmarks.