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This paper proposes a mapping strategy and a code construction method for 2m-ary polar-coded modulation. In order to find a good mapping for a polar code in an efficient way, we reduce a search space for mapping patterns by exploiting the properties of its polarizing matrix. Once a mapping is selected, the set of unfrozen information bits to define a polar code is automatically determined. Numerical results show that our approach can provide a performance gain of about 0.3-0.5 dB over a conventional approach using random mapping and the polar code optimized for binary phase-shift keying (BPSK) modulation, when pulse-amplitude modulation (PAM) with Gray labelling is employed.