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In this correspondence, we first present a transformation technique to improve the normalized diversity product for a full rate algebraic space-time block code (STBC) by balancing the signal mean powers at different transmit antennas. After rewriting a cyclic division algebra structure into a multilayer structure for a full rate code, we show that the normalized diversity product of the transformed code with the multilayer structure is better than the one of the transformed code with the cyclic division algebra structure. We then present a new full rate algebraic STBC with multilayer structure with nonvanishing determinant (NVD) for three transmit antennas when signal constellation is carved from QAM. We show that the new code has larger normalized diversity product than the existing 3 times 3 NVD full rate STBC for quadrature amplitude modulation (QAM) signals, and we also show that it has the largest normalized diversity product in a family of full rate STBC.