A note on the nonexistence of Barker sequences

A Barker sequence is a sequence with elements ±1 such that all out-of-phase aperiodic autocorrelation coefficients are 0, 1 or -1. It is known that if a Barker sequence of length s > 13 exists then s = 4N² for some odd integer N 55, and it has long been conjectured that no such sequence exists. We review some previous attempts to improve the bound on N which, unfortunately, contain errors. We show that a recent theorem of Eliahou et al. [5] rules out all but six values of N less than 5000, the smallest of which is 689.