Three-Phase Barker Arrays

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 x 2 or 3 x 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a doubleexponentially growing arithmetic function T such that no 3-phase Barker array of size s x t with 3 vertical bar t exists for all t < T(s). For example, T(5) = 4860, T(10) > 10(11), and T(20) > 10(214). When both dimensions are divisible by 3, the existence problem is settled completely: if a 3-phase Barker array of size 3r x 3q exists, then r = q = 1. (C) 2013 Wiley Periodicals, Inc.