Summary: An injective \(k\)-edge coloring of a graph \(G=(V(G),E(G))\) is a \(k\)-edge coloring \(\varphi\) such that if \(e_1\) and \(e_2\) are at distance exactly 2 or in the same triangle, then \(\varphi (e_1)\ne \varphi (e_2)\). The injective chromatic index of \(G\), denoted by \(\chi_i'(G)\), is the minimum \(k\) such that \(G\) has an injective \(k\)-edge coloring. The edge weight of \(G\) is defined as \(ew(G)=\max \{d_G(u)+d_G(v):uv\in E(G)\}\). In this paper, we show that \(\chi_i^\prime(G)\le 3\) if \(ew(G)\le 5; \chi_i^\prime(G)\le 7\) if \(ew(G)\le 6\); and \(\chi_i^\prime(G)\le 11\) if \(ew(G)\le 7\).