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Goethals–Seidel Difference Families with Symmetric or Skew Base Blocks


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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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).

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