Finite Fields package seemed very useless for my purpose.

Is there a way to do some linear algebra computation (finding rank, solving linear systems, finding span of vectors) over a finite field (GF(2), GF(5), GF(3^5) etc) ?

Thank you in advance

I want to construct a list/set of square symmetric matrices (or Toeplitz etc.) matrices with entries from a certain finite field, GF(2) or GF(5).  

Thank you in adavnce.

How can I define a set generated by two degree n polynomials with binary coefficients?

For example: for n=2, I want to generate the set {(a_0*x_0+a_1*x_1)* (b_0*y_0+b_1*y_1) s.t. a_0,a_1,b_0,b_1 are binary}. 

Thank you in advance.