Counterexamples to Theorem 1 of Turyn's and Storer's Paper "On Binary Sequences"

Abstract

Explicit counterexamples to Theorem 1 of R. Turyn's and J. Storer's often-cited paper "On Binary Sequences" [Proc. Amer. Math. Soc., vol. 12, no. 3, pp. 394-399, 1961] are given. Theorem 2 of their paper is well known; it states that Barker sequences of odd length n>13 do not exist. Since the proof of Theorem 2 relies on the incorrect Theorem 1, the proof of the often-cited Theorem 2 is therefore not correct. It is not at all clear how Theorem 1 and its proof must be modified in order to get a correct proof of the well-known Theorem 2.