Let $Ainmathbb{R}^{mtimes k}$ be a fixed matrix with $m<k$, and let $X$ be a random vector distributed uniformly on the $k$-cube ${-1,1}^{k}$.

What is the distribution of $Y=AX$?

Denoting by $a_{i}^{top}$ the $i$th row of $A$, we can see that the $i$th entry of $Y$ can take (at most) $2^{k}$ possible values from ${a_{i1}+cdots+a_{ik},ldots,-a_{i1}-cdots-a_{ik}}$, and each of these is equally likely. How do I compute the full joint distribution? Any hints appreciated.

(Also posted on Math StackExchange)