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Goethals–Seidel Difference Families with Symmetric or Skew Base Blocks

Abstract

We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals–Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group of order v, which can be used to construct Hadamard matrices via the well-known Goethals–Seidel array. We consider the special class of these families in cyclic groups, where each base block is either symmetric or skew. We omit the well-known case where all four blocks are symmetric. By extending previous computations by several authors, we complete the classification of GS-difference families of this type for odd \(v<50\). In particular, we have constructed the first examples of so called good matrices, G-matrices and best matrices of order 43, and good matrices and G-matrices of order 45. We also point out some errors in one of the cited references.

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Acknowledgements

The authors would like to thank the referees for their constructive comments, that contributed to improving the paper. The authors wish to acknowledge generous support by NSERC, Grant Numbers 5285-2012 and 213992. Computations were performed on the SOSCIP Consortium’s Blue Gene/Q, computing platform. SOSCIP is funded by the Federal Economic Development Agency of Southern Ontario, IBM Canada Ltd., Ontario Centres of Excellence, Mitacs and 14 academic member institutions.

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Correspondence to Dragomir Ž. Ɖoković.

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Ɖoković, D.Ž., Kotsireas, I.S. Goethals–Seidel Difference Families with Symmetric or Skew Base Blocks. Math.Comput.Sci. 12, 373–388 (2018). https://doi.org/10.1007/s11786-018-0381-1

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  • DOI: https://doi.org/10.1007/s11786-018-0381-1

Keywords

  • Goethals–Seidel array
  • Difference families
  • Good matrices
  • G-matrices
  • Best matrices

Mathematics Subject Classification

  • 05B10
  • 05B20