Given a $k$-uniform hypergraph $H$ on $n$ vertices, an even cover in $H$ is a collection of hyperedges that touch each vertex an even number of times. Even covers are a generalization of cycles in graphs and are equivalent to linearly dependent subsets of a system of linear equations modulo $2$. As a result, they arise naturally in the context of well-studied questions in coding theory and refuting unsatisfiable $k$-SAT formulas. Analogous to the irregular Moore bound of Alon, Hoory, and Linial (2002), in 2008, Feige conjectured an extremal trade-off between the number of hyperedges and the length of the smallest even cover in a $k$-uniform hypergraph. This conjecture was recently settled up to a multiplicative logarithmic factor in the number of hyperedges (Guruswami, Kothari, and 1Manohar 2022 and Hsieh, Kothari, and Mohanty 2023). These works introduce the new technique that relates hypergraph even covers to cycles in the associated \emph{Kikuchi} graphs. Their analysis of these Kikuchi graphs, especially for odd $k$, is rather involved and relies on matrix concentration inequalities. In this work, we give a simple and purely combinatorial argument that recovers the best-known bound for Feige's conjecture for even $k$. We also introduce a novel variant of a Kikuchi graph which together with this argument improves the logarithmic factor in the best-known bounds for odd $k$. As an application of our ideas, we also give a purely combinatorial proof of the improved lower bounds (Alrabiah, Guruswami, Kothari and Manohar, 2023) on 3-query binary linear locally decodable codes.

翻译：给定一个$n$顶点上的$k$一致超图$H$，$H$中的偶覆盖是满足每个顶点被覆盖偶数次的超边集合。偶覆盖是图中圈的推广，等价于模2线性方程组中的线性相关子集。因此，它们自然出现在编码理论和反驳不可满足$k$-SAT公式中一些被广泛研究的问题中。类似于Alon、Hoory和Linial（2002）的不规则摩尔界，Feige在2008年提出了一个关于$k$一致超图中超边数量与最小偶覆盖长度之间的极值权衡猜想。该猜想最近被解决至超边数量的乘法对数因子内（Guruswami、Kothari和Manohar 2022；Hsieh、Kothari和Mohanty 2023）。这些工作引入了将超图偶覆盖与相关菊池图中的圈联系起来的新技术。他们对这些菊池图（尤其当$k$为奇数时）的分析相当复杂，且依赖于矩阵集中不等式。在这项工作中，我们给出一个简单且纯组合的论证，恢复了Feige猜想在偶数$k$情况下的最佳已知界。我们还引入了一种新颖的菊池图变体，结合这一论证改进了奇数$k$最佳已知界中的对数因子。作为我们思想的应用，我们还给出了关于3查询二元线性局部可解码码的改进下界（Alrabiah、Guruswami、Kothari和Manohar，2023）的纯组合证明。