On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Abstract

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.