Contract grant sponsor: NSERC (to J.B and J.J ); contract grant sponsor: German Research Foundation (to K.-U.S).
Three-Phase Barker Arrays
Article first published online: 20 OCT 2013
© 2013 Wiley Periodicals, Inc.
Journal of Combinatorial Designs
How to Cite
Bell, J. P., Jedwab, J., Khatirinejad, M. and Schmidt, K.-U. (2013), Three-Phase Barker Arrays. J. Combin. Designs. doi: 10.1002/jcd.21377
- Article first published online: 20 OCT 2013
- Manuscript Revised: 18 SEP 2013
- Manuscript Received: 1 JUN 2013
- German Research Foundation
- Barker array;
- aperiodic autocorrelation;
- algebraic number theory
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 with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3-phase Barker array of size exists, then .