This paper concerns with efficient communication over Gaussian and fading multiple-access channels (MACs). Existing orthogonal multiple-access (OMA) and power-domain nonorthogonal-OMA (NOMA) cannot achieve all rate-tuples in the MAC capacity region. Meanwhile, code-domain NOMA schemes usually require big-loop receiver-iterations for multi-user decoding, which is subject to high implementation cost and latency. This paper studies a linear physical-layer network coding multiple access (LPNC-MA) scheme that is capable of achieving any rate-tuples in the MAC capacity region without receiver iterations. For deterministic Gaussian MACs with $M$ users, we propose to utilize $q$ -ary irregular repeat accumulate (IRA) codes over finite integer fields/rings and $q$ -ary pulse amplitude modulation ( $q$ -PAM) as the underlying coded-modulation. The receiver sequentially computes $M$ network coded (NC) messages of the $M$ users. All users’ messages are then recovered by solving the computed $M$ NC messages via the inverse of the NC coefficient matrix. A joint nested code construction and extrinsic information transfer (EXIT) chart based code optimization method is developed, yielding near-capacity performance (within 0.7 and 1.1 dB the capacity limits for two and three users respectively). For fading MAC, we study the symmetric rate of LPNC-MA, and propose a pragmatic method for identifying the mutual information (MI) maximizing network coding coefficient matrix. Numerical results demonstrate that the frame error rate (FER) of the optimized LPNC-MA is within a fraction of dB the outage probability of fading MAC capacity. LPNC-MA remarkably outperforms NOMA-SIC and IDMA while avoiding the big-loop receiver iteration.