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Keywords:

  • Barker array;
  • aperiodic autocorrelation;
  • three-phase;
  • algebraic number theory

Abstract

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 inline image with inline image exists for all inline image. For example, inline image, inline image, and inline image. When both dimensions are divisible by 3, the existence problem is settled completely: if a 3-phase Barker array of size inline image exists, then inline image.