  We complete the classification of tight $4$-designs in Hamming association
schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which
had been open since a result by Noda (1979). To do so, we construct an
association scheme attached to a tight $4$-design in $H(n,q)$ and analyze its
triple intersection numbers to conclude the non-existence in all open cases.
