It is shown how to construct a general linear hybrid cellular automaton (CA) such that it has a maximum length cycle, and how the aliasing properties of such automata compare with linear feedback shift registers (LFSRs) when used as signature analyzers. The construction is accomplished by formally demonstrating the isomorphism which binds this kind of CA to the LFSRs. Consequently, these CAs can be analyzed as linear machines. Linear algebraic techniques are then applied appropriately for the transformations, and a useful search algorithm is developed which, given an irreducible characteristic polynomial, finds a corresponding linear hybrid automaton. Such CAs are tabulated for all irreducible and primitive polynomials up to degree 16, plus a selection of others of higher degree. The behavior of a linear hybrid CA and that of its corresponding LFSR are similar-that is, they have the same cycle structure and only relabel the states. The aliasing properties, when they are used as signature analyzers, remain unchanged.>