Arithmetic correlation of binary half-ℓ-sequences

The arithmetic correlations of two binary half-ℓ-sequences with connection integer p^r, which is an odd prime power, are investigated. Possible values (of the arithmetic correlation) are calculated. In particular, if p ≡ 1 (mod 8), the authors prove that they are zero for non-trivial shifts, that is, the half-ℓ-sequences have ideal arithmetic correlations. If p ≡ −1 (mod 8), an upper bound, which is of order of magnitude p^r−1/2 ln p, is derived by using earlier results on the imbalance of half-ℓ-sequences with connection integer p studied by Gu and Klapper and later improved by Wang and Tan.