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A number of S-T (space-time) block codes have been proposed for the quasi-static, flat-fading channel with coherent receiver. Most of these are linear codes, i.e., the codeword is a linear function of the scalar input symbols. We propose new non-linear S-T block codes, i.e., the codewords are non-linear functions of the input symbols. We demonstrate a non-linear code that outperforms the corresponding linear code by 0.6 to 1.2 dB. We draw parallels between optimal S-T code design and the well-known simplex conjecture for multi-dimensional AWGN codes. Finally we show that for certain nonuniformly distributed input symbols, the optimal signal set cannot be designed with linear S-T modulation and non-linear modulation is essential for optimality. This has applications in the design of S-T trellis codes and concatenated codes.