existence of solutions of an over-determined system over finite fields

Question

Does anyone know how to determine whether or not a linear equation array over finite fields has at least a solution? Basically, I have xA=b, where x has length n and A has size n by m. When m>n, the system over-determined. I know if everything is over real number, the condition would be AA'b=b, where A' is the pseudo-inverse of A. But how about system over finite field? I am primarily using MATLAB, and pinv(A) doesnt work for A over finite fields.

Thanks in advance!

Answer 1

I don't have the Communications System Toolbox, so I can't try anything here out, but to check if your system for existence and/or uniqueness of solutions, try gfrank. Check if the rank of the matrix is equal to the number of rows (unique solution). Rather, compare rank([A,b]) and rank(A), although I don't know if Galois field objects concatenate like that.

Then use the backslash operator (\ a.ka. mldivide) to solve the system Ax=b.

See here for examples of linear algebra with Galois fields.