Abstract: In 2019, Junnila, Laihonen and Paris studied the identifying codes and locating codes on the circulant graphs $C_{n}(1,d)$, $C_{n}(1,d-1,d)$ and $C_{n}(1,d-1,d,d+1)$. In this paper we study the identifying codes and locating codes of Cayley graphs on the Abelian groups of order $p^{2}$ and order $2n$ within 8 degrees, determine their optimal bounds, and give some examples that reach the optimal bounds. Our results generalize multiple results on identifying and locating codes.

Key words: Identifying code, Locating code, Optimal code, Density