New nonbinary code bounds based on divisibility arguments

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