We consider the class of linear time invariant systems with m inputs (m actuators) and p outputs (p sensors). Our aim is to design a residual generator for detection and isolation of (m + p) failures, each of them occurring for a distinct actuator or sensor. We assume that only one fault is present at a time, and we restrict our attention to residual generators of a given structure. This allows us to associate particularly simple coding sets with the fault detection and isolation (FDI) system. Necessary and sufficient conditions for the existence of a solution to the problem are derived. Furthermore, the class of all residual generators of the required structure is parametrized.