Suppose we have a binary vector $f$ with dimensions $1×l$ such that each entry in the vector is generated independently with propability $q$ of being $1$. And we have a binary matrix $G$ with dimensions $l×n$ such that each entry in the matrix is generated independently with probability $p$ of being $1$. Now we take the product fG (where these calculations are all in GF(2), i.e. $1+1=0$) which is a row vector $x$ of dimension $1×n$.
My question is: what is the probability distribution of the output vector $x$? Are the entries in $x$ independent of one another (as far as i can tell they are not)? In the case of a fixed $f$ and random $G$ the output entries are independent and hence it is easy to calculate the probability distribution over the output binary vector, but for random $f$, I am stuck.
Thank you in advance.