NS SATURATED AND Δ1-DEFINABLE

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal M-1 , there is a model of ZFC in which NS omega 1 is aleph(2)-saturated and Delta(1)-definable with omega(1) as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with NS omega 1 saturated and Delta(1)-definable with a ladder system (C) over right arrow and a full Suslin tree T as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article.