How can the Stern's algorithm be used to attack McEliece?
Forgive me if I missed something.
Is $K$ a generator matrix? I believe yes, which means the original code is linear.
The translate of the original code by subtracting $y$ is a coset, sometimes called affine subspace. It has almost all the nice properties of a subspace.
In particular sums of elements of the translated code are codewords in the original code (characteristic two means sum is the same as difference).