  Let $ n \ge 2$ be an integer. We give necessary and sufficient conditions for
an integral quadratic form over dyadic local fields to be $ n $-universal by
using invariants from Beli's theory of bases of norm generators. Also, we
provide a minimal set for testing $ n $-universal quadratic forms over dyadic
local fields, as an analogue of Bhargava and Hanke's 290-theorem (or Conway and
Schneeberger's 15-theorem) on universal quadratic forms with integer
coefficients.
