Edge-coloring a graph $G$ so that every copy of a graph $H$ has an odd color class

Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_H(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_H(n)=D_H(n)\mathrm{/}{2}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}$. Alon observed that a lower bound on $d_H(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class. Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd color class. We prove $g(K_n,K_5)\le n^{o(1)}$ and $g(K_{n,n},C_4)=n\mathrm{/}2+o(n)$. The first result shows $d_{K_5}(n)\ge \frac{1}{n^{o(1)}}$ and was obtained independently in arXiv:2306.14682.