2017 Fiscal Year Final Research Report
Study on higher dimensional dual hyperovals and related functions on finite fields
| Project/Area Number |
26400029
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kagawa National College of Technology |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | 代数学 / 代数的組合せ論 / 有限幾何学 / 高次元双対超卵形 / APN関数 |
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| Outline of Final Research Achievements |
We discovered a family of dual hyperovals S_c(l,GF(2r)), where l, r integers and c an element of the field GF(2r). Using these dual hyperovals S_c(l,GF(2r)), we construct many simply connected examples which are not known before. We describe the cover and quotient relations among them using elements c and the integers l and r, and determined the automorphism groups of them. Next we construct dual hyperovals using a commutative presemified and determined the isomorphism problems among them. By extending this idea, we also construct dual hyperovals using many presemifields which may not be commutative. We determined the isomorphism problems among them under the specific conditions, such as the presemifields are not isotopic to commutative presemifields. Lastly, we construct a quadratic APN function using a bent function of special type. Only two examples of such constructions were known before. It is proved that our construction are not equivalent to the former two examples.
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| Free Research Field |
代数的組合せ論
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