  In this paper a wide family of identifying codes over regular Cayley graphs
of degree four which are built over finite Abelian groups is presented. Some of
the codes in this construction are also perfect. The graphs considered include
some well-known graphs such as tori, twisted tori and Kronecker products of two
cycles. Therefore, the codes can be used for identification in these graphs.
Finally, an example of how these codes can be applied for adaptive
identification over these graphs is presented.
