Identifying codes of cycles

In this paper we deal with identifying codes in cycles. We show that for all r ≥ 1 , any r -identifying code of the cycle C n has cardinality at least gcd ( 2 r + 1 , n ) ⌈ n 2 gcd ( 2 r + 1 , n ) ⌉ . This lower bound is enough to solve the case n 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 n odd seems to be more complicated. An upper bound is given for the case n odd, and some special cases are solved. Furthermore, we give some conditions on n and r to attain the lower bound.