  In this paper we consider a channel model that is often used to describe the
mobile wireless scenario: multiple-antenna additive white Gaussian noise
channels subject to random (fading) gain with full channel state information at
the receiver. Dynamics of the fading process are approximated by a
piecewise-constant process (frequency non-selective isotropic block fading).
This work addresses the finite blocklength fundamental limits of this channel
model. Specifically, we give a formula for the channel dispersion -- a quantity
governing the delay required to achieve capacity. Multiplicative nature of the
fading disturbance leads to a number of interesting technical difficulties that
required us to enhance traditional methods for finding channel dispersion.
Alas, one difficulty remains: the converse (impossibility) part of our result
holds under an extra constraint on the growth of the peak-power with
blocklength.
  Our results demonstrate, for example, that while capacities of $n_t\times
n_r$ and $n_r \times n_t$ antenna configurations coincide (under fixed received
power), the coding delay can be quite sensitive to this switch. For example, at
the received SNR of $20$ dB the $16\times 100$ system achieves capacity with
codes of length (delay) which is only $60\%$ of the length required for the
$100\times 16$ system. Another interesting implication is that for the MISO
channel, the dispersion-optimal coding schemes require employing orthogonal
designs such as Alamouti's scheme -- a surprising observation considering the
fact that Alamouti's scheme was designed for reducing demodulation errors, not
improving coding rate. Finding these dispersion-optimal coding schemes
naturally gives a criteria for producing orthogonal design-like inputs in
dimensions where orthogonal designs do not exist.
