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We develop a new framework to analyze the average block error rate (BLER) of using short-packet communications in a non-orthogonal multiple access (NOMA) system. In this system, the N-antenna base station (BS) designs its transmission weight as per the instantaneous channel knowledge from the BS to either the near user or the far user. Incorporating the unique relationship of the finite blocklength and the achievable rate from the BS to two users, we derive new closed-form expressions for the average BLER for arbitrary signal-to-noise ratios (SNRs). We also derive simple expressions for the asymptotic average BLER at high SNRs, based on which we determine the optimal power allocation and minimum blocklength for satisfying the average BLER targets at two users. We demonstrate that the user, whose channel knowledge is used for weight design, achieves a lower BLER than the other. We further show …