In this paper we construct low ML 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 F4 domain. The problem of constructing low ML decoding complexity STBCs is shown to be equivalent to finding certain codes over F 4. It is shown that almost all known low ML decoding complexity STBCs can be obtained by this approach. New classes of codes are given that have the least known ML decoding complexity in some ranges of rate. © 2010 IEEE.