Suppose we wish to transmit a vector f \varepsilon R^n reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are corrupted in a completely arbitrary fashion by an error e. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover f exactly from the corrupted m-dimensional vector y^1 = {\rm A}f + e? This paper proves that under suitable conditions on the coding matrix A, the input f is the unique solution to the (||x||\ell 1: = \Sigma i|xi|) \min ||y - {\rm A}f||\ell 1f\varepsilon R^n provided that the fraction of corrupted entries is not too large, i.e. does not exceed some strictly positive constant p^ * ( numerical values for p^ * are actually given). In other words, f can be recovered exactly by solving a simple convex optimization problem; in fact, a linear program. We report on numerical experiments suggesting that \ell 1 minimization is amazingly effective; f is recovered exactly even in situations where a very significant fraction of the output is corrupted. In the case when the measurement matrix A is Gaussian, the problem is equivalent to that of counting lowdimensional facets of a convex polytope, and in particular of a random section of the unit cube. In this case we can strengthen the results somewhat by using a geometric functional analysis approach.