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In this paper, we propose a non-binary belief propagation approach (NB-BP)for detection of $M$-ary modulation symbols and decoding of $q$-ary LDPC codesin large-scale multiuser MIMO systems. We first propose a message passing basedsymbol detection algorithm which computes vector messages using a scalarGaussian approximation of interference, which results in a total complexity ofjust $O(KN\sqrt{M})$, where $K$ is the number of uplink users and $N$ is thenumber of base station (BS) antennas. The proposed NB-BP detector does not needto do a matrix inversion, which gives a complexity advantage over MMSEdetection. We then design optimized $q$-ary LDPC codes by matching the EXITcharts of the proposed detector and the LDPC decoder. Simulation results showthat the proposed NB-BP detection-decoding approach using the optimized LDPCcodes achieve significantly better performance (by about 1 dB to 7 dB at$10^{-5}$ coded BER for various system loading factors with number of usersranging from 16 to 128 and number of BS antennas fixed at 128) compared tousing linear detectors (e.g., MMSE detector) and off-the-shelf $q$-aryirregular LDPC codes. Also, even with estimated channel knowledge (e.g., withMMSE channel estimate), the performance of the proposed NB-BP detector isbetter than that of the MMSE detector.

M-ary Detection and q-ary Decoding in Large-Scale MIMO: A Non-Binary Belief Propagation Approach