Asymptotically-Optimal, Fast-Decodable, Full-Diversity STBCs

For a family/sequence of Space-Time Block Codes (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), $i=1,2,\dots$, the \emph{asymptotic normalized rate} is defined as $\lim_{i \to \infty}{\frac{R_i}{N_i}}.$ A family of STBCs is said to be \emph{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 \emph{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 maximum-likelihood (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) Construction of a new class of asymptotically-good, full-diversity multigroup ML decodable codes, that not only includes STBCs for a larger set of antennas, but also either matches in rate or contains as a proper subset all other high-rate or asymptotically-good, delay-optimal, multigroup ML decodable codes available in the literature. (ii) Construction of a new class of fast-group-decodable codes (codes that combine the low ML decoding complexity properties of multigroup ML decodable codes and fast-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.