Block invariance and reversibility of one dimensional linear cellular automata

Consider a one-dimensional, binary cellular automaton f (the CA rule), where its n nodes are updated according to a deterministic block update (blocks that group all the nodes and such that its order is given by the order of the blocks from left to right and nodes inside a block are updated synchronously). A CA rule is block invariant over a family F of block updates if its set of periodic points does not change, whatever the block update of F is considered. In this work, we study the block invariance of linear CA rules by means of the property of reversibility of the automaton because such a property implies that every configuration has a unique predecessor, so, it is periodic. Specifically, we extend the study of reversibility done for the Wolfram elementary CA rules 90 and 150 as well as, we analyze the reversibility of linear rules with neighbourhood radius 2 by using matrix algebra techniques.