2007 Fiscal Year Final Research Report Summary
A research on higher dimensional dual hyperovals in projective spaces
| Project/Area Number |
17540054
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Takuma National College of Technology |
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Principal Investigator |
TANIGUCHI Hiroaki Takuma National College of Technology, department of general education, professor (60370037)
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| Project Period (FY) |
2005 – 2007
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| Keywords | finite geometry / finite protective spaces / dual hyperoval |
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| Research Abstract |
We investigate on higher dimensional dual hyperovals, that is, construct some new dual hyperovals and study their properties, as follows.
1. We generalize the construction of Yoshiara e give a construction of dual hyperovals in PG (n, q) with q even, and study on the isomorphism classes in case n=2d+1.
2. (Study on Buratti-Del Fra's dual hyperovals.)
(1) We realize Buratti-Del Fra's dual hyperovals in $G (3d-1, 2), which are originally constructed in PG (d (d+3)/ 2, 2). (2) We also give a characterization of Hybrechts's dual hyperovals and Buratti-Del Fra's dual hyperovals.
3. We constructed a new family of dual hyperovals in $PG (d (d+3)/2, 2)$, and study on the automorphism groups.
4. We study on d-dimensional dual hyperovals in PG (2d, 2).
(1) We construct dual hyperovals from affine translation planes of characteristic 2, and prove that some dual hyperovals constructed by us are not isomorphic to the Yoshiara's dual hyperovals.
(2) We prove that dual hyperovals constructed from Near field of characteristic 2 (by the author) are not isomorphic if Near fields are not isomorphic. Hence we prove that, for some d, there are a lot of non-isomorphic dual hyperovals in PG (2d, 2).
5. We try to generalize the definition of dual hyperovals in PG (n, q) with q odd, and give some examples.
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