Arithmetic Autocorrelation of Binary $m$-Sequences

An $m$-sequence is the one of the largest period among those produced by a linear feedback shift register. It possesses several desirable features of pseudorandomness such as balance, uniform pattern distribution and ideal autocorrelation for applications to communications. However, it also possesses undesirable features such as low linear complexity. Here we prove a nontrivial upper bound on its arithmetic autocorrelation, another figure of merit introduced by Mandelbaum for error-correcting codes and later investigated by Goresky and Klapper for FCSRs. The upper bound is close to half of the period and hence rather large, which gives an undesirable feature.