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

Arithmetic Autocorrelation of Binary $m$-Sequences