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The authors consider the following simple encoding scenario, which they term successive encoding: φx performs lossless encoding of X, at rate H(X), and φy is designed assuming that the decoder has recovered X. For coding with vanishing error probability, the results of Slepian and Wolf (1973) show that time-sharing of the two possible successive encoding scenarios (obtained by interchanging the roles of X and Y) is optimal, in that all achievable rate pairs can be achieved via this strategy. But time-sharing combined with successive encoding may not be an optimal strategy in the case of zero-error coding. The authors construct a simple example for this suboptimality phenomenon.