Soft Solution of Noisy Linear GF(2) Equations

We extend the idea of solving noisy overdetermined equations to $GF(2)$ equations. The equation $Ax = d$ is to be solved, where the right-hand side d is not known exactly, but has probabilistic errors that are characterized by a log-likelihood function. A solution is obtained by selecting full-rank submatrices and solving the resulting system of equations using tanh rule to produce multiple soft solutions, whose log likelihood functions are averaged together. Several different methods of selecting the submatrices are described. A pseudoinverse-like solution is also presented. Hard solutions are also computed. The methods are compared against each other. The soft solutions provide significant improvement compared with the hard solutions, although still not achieving maximum likelihood performance.