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In contrast to other spread-spectrum techniques, wideband pulse trains with relatively low pulse arrival rates may be considered unsuitable for covert communications. The high crest factor of such trains can be extremely burdensome for the transmitter hardware, and it makes the pulse trains easily detectable even at very low signal-to-noise ratios. In addition, it may appear that sharing the wideband channel by multiple users would require explicit allocation of the pulse arrival times for each sub-channel, which would be impractical in most cases. On the other hand, messaging by wideband pulse trains has many appealing features. Among those are the ease of synchronous as well as asynchronous pulse detection, and on-the-fly channel reconfigurability (e.g. changing the spreading factor). Favorably, the crest factor of a pulse train, as well as its apparent temporal and amplitude structure, can be easily, and reversibly, controlled by simple linear filtering. For example, a transmitted pulse train can be made statistically indistinguishable from the Gaussian component of the channel noise (e.g. the thermal noise) observed in the same spectral band, while the received signal will be the designed high-crest-factor wideband pulse train. In this paper, we utilize the so-called pulse pileup effect to perform such reversible control of the pulse train structure, enabling a wider use of this approach for synthesis of robust low-SNR covert communication links. We place a particular focus on the synchronous pulse detection in the receiver, that provides a better utilization of the channel spectrum.