The problem of decoding can be thought of as consisting of solving an ill-posed, linear inverse problem with noisy data and box constraints upon the unknowns. Specificially, we aimed to solve $\mathit{A}\mathit{x}+\mathit{e}=\mathit{y},$ where $\mathit{A}$ is a matrix with positive entries and $\mathit{y}$ is a vector with positive entries. It is required that $\mathit{x}\in \mathcal{K}$, which is specified below, and we considered two points of view about the noise term, both of which were implied as unknowns to be determined. On the one hand, the error can be thought of as a confounding error, intentionally added to the coded message. On the other hand, we may think of the error as a true additive transmission-measurement error. We solved the problem by minimizing an entropy of the Fermi-Dirac type defined on the set of all constraints of the problem. Our approach provided a consistent way to recover the message and the noise from the measurements. In an example with a generator code matrix of the Reed-Solomon type, we examined the two points of view about the noise. As our approach enabled us to recursively decrease the ${\ell }_1$ norm of the noise as part of the solution procedure, we saw that, if the required norm of the noise was too small, the message was not well recovered. Our work falls within the general class of near-optimal signal recovery line of work. We also studied the case with Gaussian random matrices.

Here we provide an entropic method to invert the Fourier transform of a bounded function defined on an interval, when the data consists of its sine and cosine transforms, but not necessarily of consecutive frequencies. The classical direct approach consists in just forming the linear combination of the trigonometric functions of the given frequencies multiplied by their respective coefficients. The problem is that this approach does not yield any information about the projection of the function on the space spanned by the missing frequencies. Our approach consists of regarding the Fourier inversion as an ill-posed linear inverse problem with box constraints, consisting of finding a function given a few of its sine and cosine transforms. To solve this problem, we propose a non-linear approach, consisting of minimizing an entropy function subject to the Fourier data as constraints. This approach provides us with a solution that has a non-vanishing projection on the space spanned by the Fourier coefficients in the original data set, from which a better approximation to the unknown function can be recovered. In addition to obtaining an explicit representation of the solution, we prove that the solution converges to the unknown function as the number of data points increases. Even though the reconstruction procedure is non-linear in the data, there is some quasi-linearity in the procedure.