  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 $\NS$
is $\aleph_2$-saturated and $\Delta_1$-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 $\NS$ saturated and $\Delta_1$-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.
