Given a ground set $N=(a_0,a_1,\dots,a_{K-1},b_0,b_1,\dots,b_{K-1})$ of size $2K$, what is the smallest matroid for which $A=(a_0,a_1,\dots,a_{K-1})$ and $B=(b_0,b_1,\dots,b_{K-1})$ are independent sets. My intuition says it might be matroids of the form: $$\mathcal{M}_t = (N,\mathcal{I}_t), \text{ where } \mathcal{I}_t = \{ S : S \cap \{ a_i,b_{i+t} \}^* \le 1 \}$$ but I am not able to prove/disprove this. Any thoughts on how I can approach this?
$*$ Indices are taken modulo $K$