  In the landmark paper by Hassibi and Hochwald, it is claimed without proof
that the upper triangular matrix R encountered during the sphere decoding of
any linear dispersion code is full-ranked whenever the rate of the code is less
than 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 at
least as much as the number of transmit antennas. We also show that all known
families of high rate (rate greater than 1 complex symbol per channel use)
multigroup ML decodable codes have rank-deficient R matrix even when the
criterion on rate is satisfied, and that this rank-deficiency problem arises
only in asymmetric MIMO with number of receive antennas less than the number of
transmit antennas. Unlike the codes with full-rank R matrix, the average sphere
decoding complexity of the STBCs whose R matrix is rank-deficient is polynomial
in the constellation size. We derive the sphere decoding complexity of most of
the 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.
