Recently in, a framework was given to construct low ML decoding complexity Space-Time Block Codes (STBCs) via codes over the finite field F₄. In this paper, we construct new full-diversity STBCs with cubic shaping property and low ML decoding complexity via codes over F₄ for number of transmit antennas N = 2^m, m >; 1, and rates R >; 1 complex symbols per channel use. The new codes have the least ML decoding complexity among all known codes for a large set of (N, R) pairs. The new full-rate codes of this paper (R = N) are not only information-lossless and fully diverse but also have the least known ML decoding complexity in the literature. For N ≥ 4, the new full-rate codes are the first instances of full-diversity, information-lossless STBCs with low ML decoding complexity. We also give a sufficient condition for STBCs obtainable from codes over F₄ to have cubic shaping property, and a sufficient condition for any design to give rise to a full-diversity STBC when the symbols are encoded using rotated square QAM constellations.