Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

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Let G=Cn1×⋯×Cnm be an abelian group of order n= n1⋯ nm, where each Cnt is cyclic of order nt. We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in G× Q8 relative to the centre Z(Q8) and the perfect arrays of size n1× ⋯ × nm over the quaternionic alphabet Q8∪ qQ8, where q= (1 + i+ j+ k) / 2. In view of this connection, for m= 2 we introduce new families of relative difference sets in G× Q8, as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.

Original languageEnglish
Pages (from-to)397–406
Number of pages10
JournalMathematics in Computer Science
Issue number4
Publication statusPublished - Dec 2018


  • Hadamard matrices
  • Perfect arrays
  • Quaternions
  • Relative difference sets

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