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<g ) for this maximum, it is shown that exᵣ ( N , C< 5 ) = Ωᵣ ( N3/2 - 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 $6$.