The authors reconsider and advance the analysis of controllability and observability (and the weaker stabilisability and detectability properties) of a class of linear networked control systems (NCSs). The authors model the NCS as a periodic system with limited communication where the non-updated signals can either be held constant (the zero-order-hold case) or reset to zero. Periodicity is dealt with the lifting technique. The authors provide conditions for controllability (stabilisability) and observability (detectability) of the NCS, given a communication sequence and the controlled plant model. These conditions allow to find communication sequences which are shorter than previously established. A strict lower bound for the sequence length is given. In the sampled-data case, the authors prove that a communication sequence that avoids particularly defined pathological sampling rates and particular eigenvalues can preserve stabilisability (and detectability for the dual problem) with a `minimum' sequence length.