On $n$-universal quadratic forms over dyadic local fields

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.