  For a graph $G$ and an integer $k\geq 2$, a $\chi'_{k}$-coloring of $G$ is an
edge coloring of $G$ such that the subgraph induced by the edges of each color
has all degrees congruent to $1 ~ (\moddd k)$, and $\chi'_{k}(G)$ is the
minimum number of colors in a $\chi'_{k}$-coloring of $G$. In ["The mod $k$
chromatic index of graphs is $O(k)$", J. Graph Theory. 2023; 102: 197-200],
Botler, Colucci and Kohayakawa proved that $\chi'_{k}(G)\leq 198k-101$ for
every graph $G$. In this paper, we show that $\chi'_{k}(G) \leq 177k-93$.
