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Unlike the half-duplex relay, the performance of the full-duplex relay is highly sensitive to the correlation between the source and relay codebooks. Linear coding complicates the design of correlated codebooks, for example in multilevel coding (MLC) linear codes at each layer can only have correlation zero or one, leading to a performance penalty that has been characterized in earlier work. In this paper, we propose a new design technique that significantly reduces the correlation penalty of linear codes via intelligent labeling for the modulation. The basic idea is as follows: the chain rule for mutual information, which is the backbone of MLC, is not-unique in two ways: the labeling of modulation constellation as well as the ordering of the chain rule. Our optimization at each level pushes the mutual information terms involving new information (for the relay) or beamforming information to zero or one. In effect this finds a suitable decomposition of overall correlation to a set of binary correlations at individual levels of MLC. Simulations show that point-to-point LDPC codes in combination with the proposed correlation design lead to excellent performance.