Supersaturated design is a form of fractional factorial design and has recently received much interest because of its potential in factor screening experiments. This paper mainly concerns the existing criteria for mixed-level designs, i.e. the x2(D) criterion (for design D) proposed by Yamada and Matsui (J. Statist.Plann. Inference 104 (2002) 459), the E (fNOD) criterion proposed by Fang et al. (Metrika 58 (2003b) 279) and the minimum moment aberration (MMA) proposed by Xu (Statist. Sinica 13 (2003) 691). A lower bound of x2(D) is obtained along with the sufficient and necessary condition for achieving it. The connections between x2(D) and other criteria are discussed. Especially, it is shown that x2(D) and the second power moment x2(D) are in fact equivalent to each other and hence the two criteria, x2(D) and MMA, share the same optimal supersaturated designs, which can be constructed from saturated orthogonal arrays and other supersaturated designs.