On Grassmann $4$-designs

We prove that there is a unique tight $4$-design in the real Grassmannian G2,4, up to the action of $O(4)$. Using symmetric association schemes, we also construct an N-point $4$-design in G2,4 for every even integer N≥10. For symmetric association schemes of order less than $21$, we classify the pairs of rank-three primitive idempotents defining injective maps into G2,4; their images are Grassmannian $4$-designs precisely for four schemes of order $16$.