  In this note, we consider the ill-posedness issue for the cubic nonlinear
Schr\"odinger equation (NLS) on the circle. In particular, adapting the
argument by Christ-Colliander-Tao [14] to the periodic setting, we exhibit a
norm inflation phenomenon for both the usual cubic NLS and the Wick ordered
cubic NLS for $s \leq s_\text{crit} :=- \frac 12$. We also discuss norm
inflation phenomena for general cubic fractional NLS on the circle.
