Design $X \in \mathcal{X}^{T \times M}$, $\mathcal{X} = \{ \pm 1 \}$ such that \begin{equation}\label{eq:mtraDes} X \mathbf{y}_{1} \ne X \mathbf{y}_{2}, \text{ } \forall \text{ } \mathbf{y}_{1} \ne \mathbf{y}_{2} \in \mathcal{X}^{M \times 1}. \end{equation} What is the minimum value for $T$ and how to design $X$?

The addition is defined over the real filed, i.e., $1 + (-1) = 0$, $1 + 1 =2$. Any thoughts are apprciated. Thanks in addvance.


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