Finding solution of $A\mathbf{x} = \mathbf{0}$ from rref over gf(2) [duplicate]

I'm working on the Gaussian elimination being implemented on gf(2). I have successfully reduced my 286*286 matrix into rref. Now I need to find the null space of this(please tell me how to do this effectively as I'm going to hardcode this in java). It would be a great help is someone could give me the gist of finding basis and then solution of this. Also as a extra detail I'm doing all this as a part of my attempt to implement Quadratic Sieve in java.

• Do you mean $A\mathbf{x} = \mathbf{0}$? And what is gf(2)? – The Pointer Apr 6 '20 at 17:47
• @ThePointer Yes Ax = 0. Gf(2) is galois field of 0 and 1. The matrix will only contain 0's and 1's. It utilizes 1-bit integer arithmetic instead. I'm stuck with solving this issue. Which in turn has been delaying the deadline for quadratic sieve implementation. – DumpDaCode Apr 6 '20 at 17:50
• @ThePointer: I suppose it is $\mathbf F_2$ (gf is for Galois field). – Bernard Apr 6 '20 at 17:51
• @rajiv You need to edit your question and make it clearer. – The Pointer Apr 6 '20 at 17:53
• @Bernard Ahh, ok. This is outside of my scope of knowledge. – The Pointer Apr 6 '20 at 17:54