In many types of modern communication, a message is transmitted over a noisy medium. When this is done, there is a chance that the message will be corrupted. An error-correcting code adds redundant information to the message which allows the receiver to detect and correct errors accrued during the transmission. We will study the famous Reed-Solomon code (found in QR codes, compact discs, deep space probes, ...) and investigate the limits of its error-correcting capacity. It can be shown that understanding this is related to understanding the "deep hole" problem, which is a question of determining when a received message has, in a sense, incurred the worst possible corruption. We partially resolve this in its traditional context, when the code is based on the finite field F_q or F_q^*, as well as new contexts, when it is based on a subgroup of F_q^* or the image of a Dickson polynomial. This is a new and important problem that could give insight on the true error-correcting potential of the Reed-Solomon code.

The purpose of this thesis is to use the tools of Inner Model Theory to the study of notions relative to generic embeddings induced by ideals. We seek to apply the Core Model Induction technique to obtain lower bounds in consistency strength for a specific stationary catching principle called StatCatch*(I), related to the saturation of an ideal I of omega_2. This principle involves the central notion of self-genericity in its formulation, introduced by Foreman, Magidor and Shelah. In particular, we show that assuming StatCatch*(I) (plus some additional hypothesis in the universe), we can obtain, for every finite n, an inner model with n Woodin Cardinals.