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In this paper, soft-decision (SD) decoders of permutation trellis code (PTC) with $M$-ary frequency shift keying are designed using three optimization algorithms and presented in four decoding schemes. In a concatenated code such as PTC, the Viterbi decoder for the outer convolutional code provides maximum likelihood decoding. Hence, the error correction performance is dependent on the decoding scheme used for the inner code. Due to the structure of the encoder with the modulation scheme, the channel output can be interpreted as an assignment problem. SD decoding can then be designed accordingly, using the presented, low-complexity optimization-based schemes. The bit error rate (BER) performance of the schemes are simulated in an additive white Gaussian noise (AWGN) and powerline communication (PLC) channel. The complexities of the schemes are also presented. The performance of the SD schemes are compared with the existing SD threshold detector, with BER results showing significant coding gain for certain codebooks. From the results, a reasonable trade-off between the complexity and coding gain is observed for a noisy channel such as the PLC channel.