2017/05/17 by Maarten De Boeck, De Boeck, Maarten, Geertrui Van de Voorde +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #math.CO
paper · pdf · doi:10.48550/arxiv.1705.06372
arxiv created 2017/05/17 · openalex publication_date 2017/05/17 · arxiv updated 2017/05/19 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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 = 2h, h > 3, an infinite family of elation KM-arcs of type q/8, and for q = 2h, where 4, 6, 7 | h an infinite family of KM-arcs of type q/16. Both families contain new examples of KM-arcs.