Tricky matrix multiplication problem when there are two possible solutions for x

Here is an example of what I mean:

We have

so basically,

s(1) and s(2) always equal something (1 or 0), so in one case, s(1) & s(2) = 0, and in another, s(1) = 0 & s(2) = 1 etc.

Note that I am using bitwise XOR addition

This results in two different possible x value combinations. In the case of s(1) & s(2) = 0, we get

and

What I am struggling to figure out is how I can program this operation so that given varying s(1) and s(2), I get 2 results for x.

Any advice on where to go would be great, I am somewhat new to MATLAB so I might be missing obvious operations but I can't find them. Thank you!

[John D'Errico]

So you want to do all computations in GF(2), for some general (not currently specified) linear system?

[Brett Wood]

Yeah, I guess my problem is really making it so my computations are in GF(2) thus yielding two different solutions for x

[Brett Wood]

I am able to get one solution using solution = gflineq(H, s, 2) but I only get 1 solution, not both