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This paper addresses the design of high rate quasi-orthogonal space-time block codes, with rates of 1.5 and 13/8, to give improved performance. Three different quasi-orthogonal designs are presented which are variants of previous quasi-orthogonal designs to give higher code rates. The first two variants transform rate 1 designs to rate 1.5 designs, the third variant transforms a rate 9/8 design to a rate 13/8 design. The designs presented are shown to be particularly attractive in terms of greater effective throughput for medium-to-high signal-to-noise ratio with a 4-ary input constellation.