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 $6$.

翻译：本文研究顶点数为$N$、$r$一致、围长$g$（其中$g \in \{5,6\}$）的超图的最大边数。记该最大值为$\textrm{ex}_r ( N, \mathcal{C}_{<g} )$，我们证明了对于$r \in \{4,5,6\}$，有$\textrm{ex}_r ( N , \mathcal{C}_{ < 5} ) = \Omega_r ( N^{3/2 - o(1)} )$。本文处理了文献[31]中一个未证明的论断，该论断声称Ruzsa的技术可用于证明该下界对所有$r \geq 3$成立。我们详细解释了文献[31]提出该论断时被忽视的一个主要障碍，并证明当$r\in \{4,5,6\}$时该障碍可以被克服。我们利用编码理论中的构造方法，证明了适用于所有$r \geq 3$的非平凡下界。最后，我们借助Conlon、Fox、Sudakov和Zhao的最新成果，证明了在上界约束距离为6的线性$q$元码的大小时，编码理论中的球堆积界可被改进。