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

We consider the split special orthogonal group (Formula presented.) defined over a (Formula presented.) -adic field. We determine the structure of any (Formula presented.) -packet of (Formula presented.) 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 (Formula presented.) -parameter as a representation of the Weil group of (Formula presented.). Our result is new when (Formula presented.) and our method provides a new proof even when (Formula presented.).