Soft Linear Algebra over Noisy GF(2) matrices

In this paper we describe methods of finding soft solutions to the linear equation A x=d, where A and d have GF(2) elements but the elements of A and d are only known probabilistically. The solutions described here provide probabilities on the elements of the solution x. Two solution methods are presented. The first method is similar to the LU factorization algorithm. Solution using this technique makes use of a soft inner product lemma proved here. The second method performs Gauss-Jordan reduction of hard and soft matrices, and provides only a solution.