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                <term xml:lang="en"> Linear feedback shift registers</term>
                <term xml:lang="en"> Linear recurring sequences</term>
                <term xml:lang="en"> Surjectivity</term>
                <term xml:lang="en"> Preimages</term>
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              <p>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.</p>
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