Let $C$ be a linear code of length $n$ over $\mathbb F_q$ with minimum distance $d$, and let $C^\perp$ have minimum distance $d^\perp$. Fix $t<d$, and suppose that among the nonzero weights $i$ with $i\le n-t$, the dual code $C^\perp$ has at most $d-t$ distinct weights. Then, for every weight $w$ with $A_w \ne 0$ and $d \le w \le n$, the supports of the weight-$w$ codewords of $C$, counted with codeword multiplicity and then identified when passing to the corresponding simple support design, form a $t$-design whenever the standard multiplicity-to-support reduction is uniform. The analogous conclusion holds for $C^\perp$ with the roles of $C$ and $C^\perp$ interchanged.