Upper bounds for linear graph codes

A linear graph code is a family $\mathcal{C}$ of graphs on $n$ vertices with the property that the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In this article, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph $H$. In particular, we show that if $H$ has an even number of edges, the size of the code is $O(2^{\binom{n}{2}}/\log n)$, making progress on a question of Alon. Furthermore, we show that for almost all graphs $H$ with an even number of edges, there exists $\varepsilon_H>0$ such that the size of a linear graph code without a copy of $H$ is at most $2^{\binom{n}{2}}/n^{\varepsilon_H}$.