Goethals--Seidel difference families with symmetric or skew base blocks
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866929216204832768 |
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| author | Djokovic, Dragomir Z. Kotsireas, Ilias S. |
| author_facet | Djokovic, Dragomir Z. Kotsireas, Ilias S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_00556 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Goethals--Seidel difference families with symmetric or skew base blocks Djokovic, Dragomir Z. Kotsireas, Ilias S. Combinatorics 05B10, 05B20 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. |
| title | Goethals--Seidel difference families with symmetric or skew base blocks |
| topic | Combinatorics 05B10, 05B20 |
| url | https://arxiv.org/abs/1802.00556 |