  A $k$-proper edge-coloring of a graph G is called adjacent
vertex-distinguishing if any two adjacent vertices are distinguished by the set
of colors appearing in the edges incident to each vertex. The smallest value
$k$ for which $G$ admits such coloring is denoted by $\chi'_a(G)$. We prove
that $\chi'_a(G) = 2R + 1$ for most circulant graphs $C_n([1, R])$.
