# Injective matrix $X \in \{ \pm 1 \}^{T \times M}$ for any $\mathbf{y} \in \mathcal{X}^{M \times 1}$

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