2 × 2 delay-tolerant distributed space-time codes with non-vanishing determinants

Distributed space-time codes over two asynchronous relays are considered. First, we show that the space-time code proposed in [1] is suitable for asynchronous transmission over two relays. Using tools from division algebra, we study the quadratic form resulting from its determinant and we prove that this code has a non-vanishing determinant over all constellations carved from Zopf[i], and thus is optimal in the sense of diversity-multiplexing tradeoff [2]. Then, we propose a delay-tolerant code based on the Golden code [3] the full-rate full-diversity information lossless space-time code proposed for the MIMO channel, and we deduce the convenient unitary matrices to obtain the modified code. Applying these matrices to other MIMO codes, namely Tirkkonen-Hottinen [4] and Sezginer-Sari code [5], we infer new delay-tolerant codes. In addition of being suitable for asynchronous relay transmission, all the new codes have the same determinants as the old ones.