NS saturated and ${\mathrm{\Delta }}_1$-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 $\text{ZFC}$ in which $\text{NS}$ is ${\mathrm{\aleph }}_2$-saturated and ${\mathrm{\Delta }}_1$-definable with ${\omega }_1$ as a parameter which answers a question of Sy-David Friedman and Liuzhen Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\text{NS}$ saturated and ${\mathrm{\Delta }}_1$-definable with a ladder system $\vec{C}$ 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.