Solving systems of linear Boolean equations with noisy right-hand sides over the reals

The paper is concerned with the problem of solution of a system of linear equations with noisy right-hand side in the following setting: one knows a random $m\times N$-matrix $A$ with entries from $\{-1,1\}$ and a vector $xA+\xi\in \R^N$, where $\xi$ is the noise vector from $\R^N$, whose entries are independent realizations of a normally distributed random variable with parameters $0$ and $\sigma^2$, and $x$ is a random vector with coordinates from $\{-1,1\}$. The sought-for parameter is the vector $x$. We propose a method for constructing a set containing the sought-for vector with probability not smaller than the given one and estimate the cardinality of this set. Theoretical calculations of the parameters of the method are illustrated by experiments demonstrating the practical implementability of the method for cases when direct enumeration of all possible values of $x$ is unfeasible.