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Rate assignment for multi-level polarised non-binary polar codes

In this study, rate assignment for finite length non-binary polar codes for arbitrary input discrete memoryless channels (AI-DMC) which can be polarised into multiple levels is discussed. The closed form expression of the block error rate (BLER) of non-binary polar codes when transmitted over an AI-DMC is derived first. Then, an element-wise exchange (EWEX) method is proposed to perform rate assignment to non-binary polar codes aiming at minimising the BLER. It is also observed that the complexity of the EWEX method grows linearly with the size of the feasible set. The construction of finite length 4-ary polar codes for a 4-ary erasure channel which can be polarised into three levels is described in detail. Simulation results show that the codes constructed by the EWEX method have almost the same BLER performance as those constructed by the exhaustive searching method. Moreover, the estimated BLERs keep consistent with the simulation results.