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We present a unified approach to designing space-time (ST) block codes using linear constellation precoding (LCP). Our designs are based either on parameterizations of unitary matrices, or on algebraic number-theoretic constructions. With an arbitrary number of Nt transmit- and Nr receive-antennas, ST-LCP achieves rate 1 symbol/s/Hz and enjoys diversity gain as high as NtNr over (possibly correlated) quasi-static and fast fading channels. As figures of merit, we use diversity and coding gains, as well as mutual information of the underlying multiple-input-multiple-output system. We show that over quadrature-amplitude modulation and pulse-amplitude modulation, our LCP achieves the upper bound on the coding gain of all linear precoders for certain values of Nt and comes close to this upper bound for other values of Nt, in both correlated and independent fading channels. Compared with existing ST block codes adhering to an orthogonal design (ST-OD), ST-LCP offers not only better performance, but also higher mutual information for Nt>2. For decoding ST-LCP, we adopt the near-optimum sphere-decoding algorithm, as well as reduced-complexity suboptimum alternatives. Although ST-OD codes afford simpler decoding, the tradeoff between performance and rate versus complexity favors the ST-LCP codes when Nt, Nr, or the spectral efficiency of the system increase. Simulations corroborate our theoretical findings.