  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.
