Graph-codes

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $[n]=\{1,2, \ldots ,n\}$ is the graph on $[n]$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. Let $H$ be a fixed graph with an even (positive) number of edges, and let $D_H(n)$ denote the maximum possible cardinality of a family of graphs on $[n]$ containing no two members whose symmetric difference is a copy of $H$. Is it true that $D_H(n)=o(2^{n \choose 2})$ for any such $H$? We discuss this problem, compute the value of $D_H(n)$ up to a constant factor for stars and matchings, and discuss several variants of the problem including ones that have been considered in earlier work.