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Elation KM-arcs

Maarten De Boeck (UGent) and Geertrui Van de Voorde (UGent)
(2019) COMBINATORICA. 39(3). p.501-544
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Abstract
In this paper, we study KM-arcs in PG(2, q), the Desarguesian projective plane of order q. A KM-arc A of type t is a natural generalisation of a hyperoval: it is a set of q+t points in PG(2, q) such that every line of PG(2, q) meets A in 0, 2 or t points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type q/4 in PG(2, q) are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type q=4. Furthermore, we construct for all q = 2(h), h > 3, an infinite family of elation KM-arcs of type q/8, and for q=2(h), where 4, 6, 7 | h an infinite family of KM-arcs of type q/16. Both families contain new examples of KM-arcs.

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MLA
De Boeck, Maarten, and Geertrui Van de Voorde. “Elation KM-Arcs.” COMBINATORICA, vol. 39, no. 3, 2019, pp. 501–44, doi:10.1007/s00493-018-3806-1.
APA
De Boeck, M., & Van de Voorde, G. (2019). Elation KM-arcs. COMBINATORICA, 39(3), 501–544. https://doi.org/10.1007/s00493-018-3806-1
Chicago author-date
De Boeck, Maarten, and Geertrui Van de Voorde. 2019. “Elation KM-Arcs.” COMBINATORICA 39 (3): 501–44. https://doi.org/10.1007/s00493-018-3806-1.
Chicago author-date (all authors)
De Boeck, Maarten, and Geertrui Van de Voorde. 2019. “Elation KM-Arcs.” COMBINATORICA 39 (3): 501–544. doi:10.1007/s00493-018-3806-1.
Vancouver
1.
De Boeck M, Van de Voorde G. Elation KM-arcs. COMBINATORICA. 2019;39(3):501–44.
IEEE
[1]
M. De Boeck and G. Van de Voorde, “Elation KM-arcs,” COMBINATORICA, vol. 39, no. 3, pp. 501–544, 2019.
@article{8630812,
  abstract     = {{In this paper, we study KM-arcs in PG(2, q), the Desarguesian projective plane of order q. A KM-arc A of type t is a natural generalisation of a hyperoval: it is a set of q+t points in PG(2, q) such that every line of PG(2, q) meets A in 0, 2 or t points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type q/4 in PG(2, q) are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type q=4. Furthermore, we construct for all q = 2(h), h > 3, an infinite family of elation KM-arcs of type q/8, and for q=2(h), where 4, 6, 7 | h an infinite family of KM-arcs of type q/16. Both families contain new examples of KM-arcs.}},
  author       = {{De Boeck, Maarten and Van de Voorde, Geertrui}},
  issn         = {{0209-9683}},
  journal      = {{COMBINATORICA}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{501--544}},
  title        = {{Elation KM-arcs}},
  url          = {{http://doi.org/10.1007/s00493-018-3806-1}},
  volume       = {{39}},
  year         = {{2019}},
}
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