Achieving the DMD tradeoff of the MIMO-ARQ channel

For the quasi-static, Rayleigh-fading MIMO channel with nt transmit and nr receive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and multiplexing gains, referred to as the diversity-multiplexing gain (D-MG) tradeoff. Explicit constructions for D-MG optimal ST codes are now available. In a subsequent paper, El Gamal, Caire and Damen considered signaling across the quasi-static ST channel using an L-round ARQ protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information (ACK or NACK) per use of the feedback channel. They showed that given a fixed number of ARQ rounds L, there is a tradeoff between diversity and multiplexing gains under which the optimum diversity gain of the ARQ channel transmitting R = r log(SNR) bits per channel use, is that of the quasi-static channel without feedback investigated by Zheng and Tse, transmitting at (1/L)th the information rate. This tradeoff, which now is a function of the number L of ARQ rounds, is termed the diversity-multiplexing gain-delay (DMD) tradeoff. In the current paper, a sufficient condition under which a ST code will achieve the DMD tradeoff is presented for the case nr ges nt. The cyclic-division-algebra-based constructions of DMG optimal ST codes by Elia et. al. are then modified to yield codes which meet this sufficient criterion and are thereby DMD-optimal. This modification requires that either nt|L or L|nt