In this article we give: (a) a new construction for mapping binary sequences to permutation sequences formed from a 4-PAM constellation, and call the resulting codebooks mappings; (b) a metric for assessing the performance of mappings from our construction; (c) performance results comparing mappings from our construction against the conventional mappings in the literature. The results show that our mappings outperform the conventional mappings. Against conventional soft-decision decoded 4-PAM, our mapping showed 2.5 dB and 3.2 dB improvement over (R = 1/3, K = 4, dfree = 10) and (R = 1/3, K = 3, dfree = 6) convolutional codes, respectively.