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The reductions of peak-to-average power ratio (PAPR) and block error rate (BLER) are two challenges in wireless systems employing orthogonal frequency-division multiplexing (OFDM)/orthogonal frequency-division multiple access. High BLER renders the system unreliable, and high PAPR is associated with power inefficiency and nonlinearity of the system. These two issues have separately been studied in the literature, but few works have studied simultaneous reductions of PAPR and BLER. In this paper, we propose a new scheme to jointly reduce and tradeoff PAPR and BLER in OFDM systems using random network coding (NC). In our proposed scheme, different representations of the input information block are generated using a special form of NC matrices, for which we prove it achieves the minimum BLER. We then propose an additional step to our proposed scheme to tradeoff a further improvement in PAPR against degradation in BLER using encoded block puncturing. Simulation results show that the proposed scheme achieves the same PAPR as conventional selective mapping (C-SLM) schemes while achieving the minimum BLER. We also show through simulations the PAPR gains achieved by our proposed additional step over C-SLM and the tradeoff of this gain against BLER degradation. Simulations finally show that our proposed scheme achieves the same results for the recently developed cubic metric.