On the Existence Conditions for Balanced Fractional $2^{m}$ Factorial Designs of Resolution $\mathrm{R}^{\ast}(\{1\}|\mathrm{\Omega}_{\ell})$ with $N<\nu_{\ell}(m)$

We consider a fractional $2^{m}$ factorial design derived from a simple array (SA) such that the $(\ell+1)$-factor and higher-order interactions are assumed to be negligible, where $2\ell\le m$. Under these situations, if at least the main effect is estimable, then a design is said to be of resolution $\mathrm{R}^{\ast}(\{1\}|\mathrm{\Omega}_{\ell})$. In this paper, we give a necessary and sufficient condition for an SA to be a balanced fractional $2^{m}$ factorial design of resolution $\mathrm{R}^{\ast}(\{1\}|\mathrm{\Omega}_{\ell})$ for $\ell=2,3$, where the number of  assemblies is less than the number of non-negligible factorial  effects. Such a design is concretely characterized by the suffixes of the indices of an SA.