# Given a system in $\mathbb{F}_2$ in RREF, how do I find a solution of minimal norm?

I have a $$12 \times 12$$ (so not really large) system of linear equations in $$\mathbb{F}_2$$ which I got to RREF through the usual row reduction. Suppose the system has multiple solutions, and call the unknowns $$x_i$$. What is the least expensive way to find a solution that minimizes the amount of $$x_i$$'s such that $$x_i = 1$$, or equivalently, a solution of minimal norm? Is this solution unique?

• Well if it is homogenous the 0 is your answer. But if you want to ignore that then no it wont be unique, a null space with basis $\langle (1,0,1),(1,1,0)\rangle$ exists and has no nonzero vector of hamming weight 1. – Algeboy Aug 5 '19 at 4:11
• No, it is not homogenous. – Fabian Levican S. Aug 5 '19 at 5:00