Hypergraphs of girth 5 and 6 and coding theory

In this paper, we study the maximum number of edges in an N-vertex r-uniform hypergraph with girth g where g \5, 6 \. Writing exᵣ (N, C<₆) for this maximum, it is shown that exᵣ (N, C < ₅) = ᵣ (N^3/2 - o (1) ) for r \4, 5, 6 \. We address an unproved claim from 31 asserting a technique of Ruzsa can be used to show that this lower bound holds for all r 3. We carefully explain one of the main obstacles that was overlooked at the time the claim from 31 was made, and show that this obstacle can be overcome when r \4, 5, 6\. We use constructions from coding theory to prove nontrivial lower bounds that hold for all r 3. Finally, we use a recent result of Conlon, Fox, Sudakov, and Zhao to show that the sphere packing bound from coding theory may be improved when upper bounding the size of linear q-ary codes of distance 6.