Three-phase Barker arrays

Jason P. Bell*, Jonathan Jedwab, Mahdad Khatirinejad , Kai Uwe Schmidt

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)


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 doubleexponentially growing arithmetic function T such that no 3-phase Barker array of size s × t with 3

Original languageEnglish
Pages (from-to)45-59
Number of pages15
JournalJournal of Combinatorial Designs
Issue number2
Publication statusPublished - 1 Jan 2015
MoE publication typeA1 Journal article-refereed


  • Algebraic number theory
  • Aperiodic autocorrelation
  • Barker array
  • Three-phase


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