A simple proof for the lower bound of the girth of graphs $D(n,q)$

The components of the graphs $D(n,q)$ provide the best-known general lower bound for the number of edges in a graph with $n$ vertices and no cycles of length less than $g$. In this paper, we give a new, short, and simpler proof of the fact that the length of the shortest cycle appearing in $D(n,q)$ is $n+5$ when $n$ is odd, and $n+4$ when $n$ is even.