Given a two rectangular binary matrices $A$ and $B$ with dimensions $c\times a$ and $c \times b$ respectively, does there exist an invertible binary matrix C with dimensions $c \times c$ such that the total number of 1 entries in $CA$ and $C^TB$ is below a target threshold $t$?
Here we are working in $GF(2)$, where multiplication and addition are AND and XOR respectively.
What is the hardness of this problem? Are there approximation algorithms to this problem?
We know this problem is in $NP$, as a valid $C$ can be used as a certificate. Also, we know how to find $C$ when there exists the a solution for $t = a+b$, by using Gaussian Elimination.