  We provide an explicit construction of the local Langlands correspondence for
general tamely-ramified reductive p-adic groups and a class of wildly ramified
Langlands parameters. Furthermore, we verify that our construction satisfies
the expected properties of such a correspondence. More precisely, we show that
each L-packet we construct admits a parameterization in terms of the Langlands
dual group, contains a unique generic element for a fixed Whittaker datum,
satisfies the formal degree conjecture, is compatible with twists by central
and cocentral characters, provides a stable virtual character, and satisfies
the expected endoscopic character identities. Moreover, we show that in the
case of GL_n, our construction coincides with the one given by Bushnell and
Henniart. Our results suggest a general approach to the construction of the
local Langlands correspondence for tamely-ramified groups and regular
supercuspidal parameters.
