Hypergraphs of girth 5 and 6 and coding theory
<|reference_start|>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 $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N , \mathcal{C}_{ < 5} ) = \Omega_r ( N^{3/2 - o(1)} )$ for $r \in \{4,5,6 \}$. We address an unproved claim from [31] asserting a technique of Ruzsa can be used to show that this lower bound holds for all $r \geq 3$. 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 $r\in \{4,5,6\}$. We use constructions from coding theory to prove nontrivial lower bounds that hold for all $r \geq 3$. 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 $6$.<|reference_end|>
@article{haymaker2024hypergraphs,
    title={Hypergraphs of girth 5 and 6 and coding theory},
    author={Kathryn Haymaker and Michael Tait and Craig Timmons},
    journal={arXiv preprint arXiv:2404.01839},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.01839},
    primaryClass={math.CO cs.IT math.IT}
}