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Frequency-selective channels can be converted to a set of flat-fading subchannels by employing orthogonal frequency-division multiplexing (OFDM). Conventional differential encoding on each subchannel, however, suffers from loss of multipath diversity, and a very high peak-to-average power ratio (PAPR), which causes undesirable nonlinear effects. To mitigate these effects, we design a block differential encoding scheme over the subchannels that preserves multipath diversity, and in addition, results in constant modulus transmitted symbols. This property is shown to ensure that the PAPR of the continuous-time transmitted waveform is reduced by a large factor. The maximum-likelihood decoder for the proposed scheme, conditioned on the current and previous received block, is shown to have linear complexity in the number of subcarriers. The constant modulus scheme will yield good bit-error rate performance with full rate only if short blocks are used. However, one may mitigate this problem by relaxing the constant modulus requirement. We show that in a practical OFDM system, we can group the subcarriers into shorter subblocks in a certain manner, and apply the constant modulus technique to each subblock. Thus, we improve diversity at a very low decoder complexity, and at the same time, we introduce an upper bound on the discrete-time PAPR, which, in turn, may lead to appreciable reduction in continuous-time PAPR, depending on the system parameters. Finally, in situations where we can sacrifice rate, additional complex field coding may be used to exploit the multipath diversity provided by channels longer than those the simple scheme can handle.