Identifying codes of cycles

In this paper we deal with identifying codes in cycles. We show that for all r≥1r≥1, any rr-identifying code of the cycle CnCn has cardinality at least gcd(2r+1,n)⌈n2gcd(2r+1,n)⌉. This lower bound is enough to solve the case nn even (which was already solved in [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (7) (2004) 969–987]), but the case nn odd seems to be more complicated. An upper bound is given for the case nn odd, and some special cases are solved. Furthermore, we give some conditions on nn and rr to attain the lower bound.