i have a huge system of equations , a part of which is in the form

x1*y1 + x4*y2 + x3*y3 + x2*y4 = 1

x2*y1 + x1*y2 + x4*y2 + x3*y3 + x4*y3 + x2*y4 + x3*y4 = 0

x3*y1 + x2*y2 + x1*y3 + x4*y3 + x3*y4 + x4*y4 = 0

x4*y1 + x3*y2 + x2*y3 + x1*y4 + x4*y4 = 0

all operations are in GF(2).

i am going to solve these equations by converting them into a sparse matrix and then solving by gauss elimination.

i want to know how i can apply linearization to get more equations ... ??? so that the system becomes solvable.

the total no. of equations of this type is greater than 1000.