On the Periods of Spatially Periodic Preimages in Linear Bipermutive Cellular Automata

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In this paper, we investigate the periods of preimages of spatially periodic configurations in linear bipermutive cellular automata (LBCA). We first show that when the CA is only bipermutive and y is a spatially periodic configuration of period p, the periods of all preimages of y are multiples of p. We then present a connection between preimages of spatially periodic configurations of LBCA and concatenated linear recurring sequences, finding a characteristic polynomial for the latter which depends on the local rule and on the configurations. We finally devise a procedure to compute the period of a single preimage of a spatially periodic configuration y of a given LBCA, and characterise the periods of all preimages of y when the corresponding characteristic polynomial is the product of two distinct irreducible polynomials.

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