Reed-Muller Codes for Quantum Pauli and Multiple Access Channels

Abstract: Reed-Muller (RM) codes have undergone significant analytical advancements over the past decade, particularly for binary memoryless symmetric (BMS) channels. We extend the scope of RM codes development and analysis to multiple-access channels (MACs) and quantum Pauli channels, leveraging a unified approach. Specifically, we first derive the achievable rate region for RM codes on so-called Q-MACs, a class of MACs with additive correlated noise. This is achieved via a generalization of the bending and boosting arguments defined in arXiv:2304.02509. We then put forward a connection between the rate region of these QMACs and quantum RM codes designed for Pauli noise channels. This connection highlights a universality property of quantum RM codes, demonstrating their rate-optimal performance across a range of channel parameters, rather than for a single Pauli channel.