Irregular hyperovals inPG(2, 64)

Penttila, Tim; Pinneri, Ivano

doi:10.1007/bf01226860

Cited by 26 publications

(14 citation statements)

References 13 publications

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“…The herd H(C3) consists of two Segre-Bartocci ovals (see [20]) and q -1 Payne ovals [14], for q > 8. Payne [16] has shown that given an elation generalized quadrangle GQ(C) associated with a q-clan, one can construct 'new' flocks via the GQ(C).…”

Section: Known Q-clans For Q Evenmentioning

confidence: 99%

Flocks and ovals

et al. 1996

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An infinite family of q-clans, called the Subiaco q-clans, is constructed for q = 2 ~. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q2, q), ovals of PG(2, q) and translation planes of order q2 with kernel GF(q). It is also shown that a q-clan, for q = 2 ~, is equivalent to a certain configuration of q + 1 ovals of PG(2, q), called a herd.Mathematics Subject Classification (1991): Primary: 51E21, 51E20, 51E12, 51E15; Secondary: 05B25.

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Section: Known Q-clans For Q Evenmentioning

confidence: 99%

Flocks and ovals

et al. 1996

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“…In [16] another approach has been used, constructing complete arcs with the constraint of being stabilized by some particular group. In this paper a similar approach has been used trying to construct complete arcs joining the orbits of some subgroup of P L (3, q).…”

Section: Approaches To Computer Searchmentioning

confidence: 99%

“…(1, 27, 2), (1, 0, 0), (1,33,22), (1,33,11), (1,28,21), (1,7,2), (1,6,39), (1,7,1), (1,22,8), (1,28,31), (1,22,16), (1,9,19), (1,1,15), (1,12,1), (1,25,17), (1,9,30) t 2 (2, 43) = 16 : (1,19,32), (1,4,13), (0, 0, 1), (1,33,2), (1,30,19), (1,…”

Section: Appendixmentioning

confidence: 99%

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Computer search in projective planes for the sizes of complete arcs

et al. 2005

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The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2, q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group P L(3, q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q = 41, 43, 47, 49, 53, 59, 64, 71 ≤ q ≤ 809, q = 529, 625, 729, and q = 821. New lower bounds on the second largest size of a complete arc are given for q = 31, 41, 43, 47, 53, 125. Also, many new sizes of complete arcs are obtained for 31 ≤ q ≤ 167. (2000): 51E21, 51E22, 94B05. Mathematics Subject Classification

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“…Such a proof would require more information concerning the groups of the hyperovals. Here it is appropriate to note that the groups of the Cherowitzo hyperovals [4] have yet to be determined, although the partial results of O'Keefe and Thas [21] were enough for Penttila and Pinneri [35] to show that the Cherowitzo hyperovals were new.…”

Section: Introductionmentioning

confidence: 98%

A unified construction of finite geometries associated withq-clans in characteristic2

Cherowitzo¹,

O'Keefe²,

Penttila³

2003

Advg

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Flocks of Laguerre planes, generalized quadrangles, translation planes, ovals, BLTsets, and the deep connections between them, are at the core of a developing theory in the area of geometry over finite fields. Examples are rare in the case of characteristic two, and it is the purpose of this paper to contribute a fifth infinite family. The approach taken leads to a unified construction of this new family with three of the previously known infinite families, namely those satisfying a symmetry hypothesis concerning cyclic subgroups of PGLð2; qÞ. The calculation of the automorphisms of the associated generalized quadrangles is su‰cient to show that these generalized quadrangles and the associated flocks and translation planes do not belong to any previously known family.

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Irregular hyperovals inPG(2, 64)

Cited by 26 publications

References 13 publications

Flocks and ovals

Flocks and ovals

Computer search in projective planes for the sizes of complete arcs

A unified construction of finite geometries associated withq-clans in characteristic2

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