Three-Phase Barker Arrays


  • Contract grant sponsor: NSERC (to J.B and J.J ); contract grant sponsor: German Research Foundation (to K.-U.S).


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 × 2 or 3 × 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 double-exponentially growing arithmetic function T such that no 3-phase Barker array of size math formula with math formula exists for all math formula. For example, math formula, math formula, and math formula. When both dimensions are divisible by 3, the existence problem is settled completely: if a 3-phase Barker array of size math formula exists, then math formula.