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This paper presents a practical technique that uses Ping's interleave(r)-division multiple access and single-user decoding to provide uncoordinated access for a family of binary-input binary-output multiple-access channels (MACs) including the OR-MAC where users' binary transmissions are combined with the logical OR operation. Information theoretic calculations provide the achievable sum-rates and optimal ones densities for these MACs. Because the required ones densities are significantly less than 50%, new nonlinear trellis code analysis and design techniques are introduced to provide the needed codes. Union bound techniques that predict the performance of these codes are also presented. Simulation results and a working FPGA implementation verify the performance and feasibility of the proposed nonlinear codes and overall multiple access scheme.