  For a family/sequence of STBCs $\mathcal{C}_1,\mathcal{C}_2,\dots$, with
increasing number of transmit antennas $N_i$, with rates $R_i$ complex symbols
per channel use (cspcu), the asymptotic normalized rate is defined as $\lim_{i
\to \infty}{\frac{R_i}{N_i}}$. A family of STBCs is said to be
asymptotically-good if the asymptotic normalized rate is non-zero, i.e., when
the rate scales as a non-zero fraction of the number of transmit antennas, and
the family of STBCs is said to be asymptotically-optimal if the asymptotic
normalized rate is 1, which is the maximum possible value. In this paper, we
construct a new class of full-diversity STBCs that have the least ML decoding
complexity among all known codes for any number of transmit antennas $N>1$ and
rates $R>1$ cspcu. For a large set of $\left(R,N\right)$ pairs, the new codes
have lower ML decoding complexity than the codes already available in the
literature. Among the new codes, the class of full-rate codes ($R=N$) are
asymptotically-optimal and fast-decodable, and for $N>5$ have lower ML decoding
complexity than all other families of asymptotically-optimal, fast-decodable,
full-diversity STBCs available in the literature. The construction of the new
STBCs is facilitated by the following further contributions of this paper:(i)
For $g > 1$, we construct $g$-group ML-decodable codes with rates greater than
one cspcu. These codes are asymptotically-good too. For $g>2$, these are the
first instances of $g$-group ML-decodable codes with rates greater than $1$
cspcu presented in the literature. (ii) We construct a new class of
fast-group-decodable codes for all even number of transmit antennas and rates
$1 < R \leq 5/4$.(iii) Given a design with full-rank linear dispersion
matrices, we show that a full-diversity STBC can be constructed from this
design by encoding the real symbols independently using only regular PAM
constellations.
