Williamson matrices and a conjecture of Ito's
We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all...
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| Format: | Journal Article |
| Language: | English |
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2009
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| Online Access: | https://hdl.handle.net/10356/92273 http://hdl.handle.net/10220/6030 |
| _version_ | 1826119599487188992 |
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| author | Bernhard, Schmidt. |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Bernhard, Schmidt. |
| author_sort | Bernhard, Schmidt. |
| collection | NTU |
| description | We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all t of the form t=2^a\cdot 10^b \cdot 26^c \cdot m with a,b,c\ge 0, m\equiv 1\ (\mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over \Z_m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t,2,4t,2t)-difference sets in Q_{8t} for every positive integer t. We also give simpler alternative constructions for relative (4t,2,4t,2t) -difference sets in Q_{8t} for all t such that 2t-1 or 4t-1 is a prime power. Relative difference sets in Q_{8t} with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito‘s conjecture for all t\le 46. |
| first_indexed | 2024-10-01T05:02:42Z |
| format | Journal Article |
| id | ntu-10356/92273 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:02:42Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | ntu-10356/922732023-02-28T19:35:33Z Williamson matrices and a conjecture of Ito's Bernhard, Schmidt. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all t of the form t=2^a\cdot 10^b \cdot 26^c \cdot m with a,b,c\ge 0, m\equiv 1\ (\mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over \Z_m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t,2,4t,2t)-difference sets in Q_{8t} for every positive integer t. We also give simpler alternative constructions for relative (4t,2,4t,2t) -difference sets in Q_{8t} for all t such that 2t-1 or 4t-1 is a prime power. Relative difference sets in Q_{8t} with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito‘s conjecture for all t\le 46. Accepted version 2009-08-11T03:07:57Z 2019-12-06T18:20:27Z 2009-08-11T03:07:57Z 2019-12-06T18:20:27Z 1999 1999 Journal Article Bernhard, S. (1999). Williamson matrices and a conjecture of Ito's. Journal of designs codes and cryptography, 17(1-3), 61-68. 0925-1022 https://hdl.handle.net/10356/92273 http://hdl.handle.net/10220/6030 10.1023/A:1008398319853 en Journal of designs codes and cryptography. Designs codes and cryptography © copyright 1999 Springer Netherlands. The journal's website is located at http://www.springerlink.com/content/m70j6m607k1630g2. 11 p. application/pdf |
| spellingShingle | DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics Bernhard, Schmidt. Williamson matrices and a conjecture of Ito's |
| title | Williamson matrices and a conjecture of Ito's |
| title_full | Williamson matrices and a conjecture of Ito's |
| title_fullStr | Williamson matrices and a conjecture of Ito's |
| title_full_unstemmed | Williamson matrices and a conjecture of Ito's |
| title_short | Williamson matrices and a conjecture of Ito's |
| title_sort | williamson matrices and a conjecture of ito s |
| topic | DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics |
| url | https://hdl.handle.net/10356/92273 http://hdl.handle.net/10220/6030 |
| work_keys_str_mv | AT bernhardschmidt williamsonmatricesandaconjectureofitos |