Is there an efficient algorithm for finding the solution $x$ of

$b = Ax$

that minimizes the Hamming weight of $x$, where

- $A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo 2") of rank $n$,
- $n<m$, say $m=500$, $n=200$,
- $b$ is a $n$-length fixed vector over $\mathbb{F}_2$ ("a binary word"),
- $x$ is a $m$-length vector (also "a binary word").

Is there an algorithm that can efficiently find a solution that is sufficiently close to the minimum?

It would be enough if there was an efficient algorithm to find the element $z \in KerA$ that minimizes the Hamming distance between an arbitrary $x$ and $z$. (Let $x$ be a solution to the equation, then $x+z$ is the solution that minimizes the Hamming weight).