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In cooperative communication networks, the performance of distributed space-time code will be severely degraded if the timing synchronization among relay nodes is not perfect. In this letter, we propose a systematic construction of the so called distributed linear convolutional space-time code (DLCSTC) for multipath fading channels that does not require the synchronization assumption. We derive sufficient conditions on the code design such that the full cooperative and multipath diversities can be achieved under the minimum memory length constraint. Then we design DLCSTCs that both have the traceorthonormality property and achieve the full diversity. We show that the proposed codes can also achieve the full diversity for asynchronous cooperative communications with ZF, MMSE and MMSE-DFE receivers under frequency-selective channels. Finally, various numerical examples are provided to corroborate the analytical studies.