Is minimising the total number of one entries in binary matrices $CA$ and $C^TB$ NP-HARD?

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.