  In this paper we construct low decoding complexity STBCs by using the Pauli
matrices as linear dispersion matrices. In this case the Hurwitz-Radon
orthogonality condition is shown to be easily checked by transferring the
problem to $\mathbb{F}_4$ domain. The problem of constructing low decoding
complexity STBCs is shown to be equivalent to finding certain codes over
$\mathbb{F}_4$. It is shown that almost all known low complexity STBCs can be
obtained by this approach. New codes are given that have the least known
decoding complexity in particular ranges of rate.
