Simple supercuspidal L $L$ -packets of split special orthogonal groups over dyadic fields

We consider the split special orthogonal group ${SO}_N$ defined over a $p$-adic field. We determine the structure of any $L$-packet of ${SO}_N$ containing a simple supercuspidal representation (in the sense of Gross–Reeder). We also determine its endoscopic lift to a general linear group. Combined with the explicit local Langlands correspondence for simple supercuspidal representations of general linear groups, this leads us to get an explicit description of the $L$-parameter as a representation of the Weil group of $F$. Our result is new when $p=2$ and our method provides a new proof even when $p\ne 2$.