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In the landmark paper by Hassibi and Hochwald, it is claimed without proofthat the upper triangular matrix R encountered during the sphere decoding ofany linear dispersion code is full-ranked whenever the rate of the code is lessthan the minimum of the number of transmit and receive antennas. In this paper,we show that this claim is true only when the number of receive antennas is atleast as much as the number of transmit antennas. We also show that all knownfamilies of high rate (rate greater than 1 complex symbol per channel use)multigroup ML decodable codes have rank-deficient R matrix even when thecriterion on rate is satisfied, and that this rank-deficiency problem arisesonly in asymmetric MIMO with number of receive antennas less than the number oftransmit antennas. Unlike the codes with full-rank R matrix, the average spheredecoding complexity of the STBCs whose R matrix is rank-deficient is polynomialin the constellation size. We derive the sphere decoding complexity of most ofthe known high rate multigroup ML decodable codes, and show that for each code,the complexity is a decreasing function of the number of receive antennas.

On the Sphere Decoding Complexity of STBCs for Asymmetric MIMO Systems