Given signal $\beta\in\{0,1\}^p$, design matrix $X\in\{0,1\}^{n\times p}$, and observations $y\in\{0,1\}^n$, where the model is \begin{align*} y=X\beta(\mod 2)\text{, or equivalently }y_i=X_i^\top\beta(\mod 2). \end{align*} (Note that we are doing modulo arithmetic at each entry of $X\beta$.)
Can I rewrite the above model in another mathematical expression that doesn't use any modulo arithmetic or any floor/ceiling functions? Ideally I would like the output to be $y\in\{-1,+1\}^n$. So the idea is you can use other alphabets instead of 0 or 1 in $\beta$ and $X$, and instead of a (mod 2) function at each entry of $X\beta$, you are allowed to use other functions.
Is this possible? Thanks.
Based on the previous comment from ronno (thanks!), I managed to get the answer \begin{align*} y_i=\frac{1-(-1)^{X_i^\top\beta}}{2}. \end{align*}