On the nonexistence of Barker arrays and related matters

Evidence is presented to support the conclusion that there exists only one equivalence class of binary two-dimensional arrays (both dimensions greater than 1) of +1's and -1's with all out-of-phase aperiodic autocorrelation values bounded in magnitude by unity. It is proved that no such array exists when either dimension is an odd prime or when one dimension is an odd integer and the other is twice an odd integer. Further constraints on the potentially possible dimensions of these structures are explored by developing their relationship to group difference sets, complementary sequences, and quaternary Barker sequences. The results of a computer search for binary arrays which have the smallest maximum out-of-phase autocorrelation magnitude are presented, and the utility of these arrays in waveform design problems is discussed