Three‐Phase Barker Arrays |
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| Authors: | Jason P Bell Jonathan Jedwab Mahdad Khatirinejad Kai‐Uwe Schmidt |
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| Affiliation: | 1. Department of Pure Mathematics, University of Waterloo, Waterloo, Canada;2. Department of Mathematics, Simon Fraser University, Burnaby, Canada;3. Department of Mathematics, The University of British Columbia, Vancouver, Canada;4. Department of Communications and Networking, Aalto University, Aalto, Finland;5. Faculty of Mathematics, Otto‐von‐Guericke University, Universit?tsplatz 6. 2, Magdeburg, Germany |
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| 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  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  . |
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| Keywords: | Barker array aperiodic autocorrelation three‐phase algebraic number theory |
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