For q,n,d∈N, let Aq(n,d) be the maximum size of a code C⊆[q]n with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds A5(8,6)≤65, A4(11,8)≤60 and A3(16,11)≤29. These in turn imply the new upper bounds A5(9,6)≤325, A5(10,6)≤1625, A5(11,6)≤8125 and A4(12,8)≤240. Furthermore, we prove that for μ,q∈N, there is a 1–1-correspondence between symmetric (μ,q)-nets (which are certain designs) and codes C⊆[q]μq of size μq2 with minimum distance at least μq−μ. We derive the new upper bounds A4(9,6)≤120 and A4(10,6)≤480 from these ‘symmetric net’ codes.