Hanani triple packings and optimal $$q$$q-ary codes of constant weight three

The exact sizes of optimal $$q$$q-ary codes of length $$n$$n, constant weight $$w$$w and distance $$d=2w-1$$d=2w-1 have only been determined for $$q\in \{2,3\}$$q∈{2,3}, and for $$w|(q-1)n$$w|(q-1)n and $$n$$n sufficiently large. We completely determine the exact size of optimal $$q$$q-ary codes of constant weight three and minimum distance five for all $$q$$q by establishing a connection with Hanani triple packings, and settling their existence.

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