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Mukerjee (1979) introduced structure (<i>k</i>) property of a factorial design. In this article, we introduce structure (<i>k</i>₁), structure (<i>k</i>₂) and structure (<i>k</i>₁<i>k</i>₂) properties of a factorial design. We establish properties of each of these structure designs in terms of the incidence and characteristic matrices of the designs. Furthermore, we develop methods of obtaining optimal <i>R</i>-type structure (<i>k</i>) designs and show that such designs are trace, <i>A</i>- and <i>MV</i>-optimal. The proposed methodologies are easy to follow and the construction of the designs comes out in a simple form.