Small maximal partial ovoids in generalized quadrangles

2013/08/08 by Jeroen Schillewaert, Jacques Verstraëte, Schillewaert, Jeroen +2
Computer Science · Mathematics · #05 #51 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Probability (math.PR) #math.CO #math.MG #math.PR #msc:05 #msc:51

paper · pdf · doi:10.48550/arxiv.1308.1899

arxiv created 2013/08/08 · openalex publication_date 2013/08/08 · arxiv updated 2013/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A \em maximal partial ovoid of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the literature. In general, theoretical lower bounds on the size of a maximal partial ovoid in a quadrangle of order (s,t) are linear in s. In this paper, in a wide class of quadrangles of order (s,t) we give a construction of a maximal partial ovoid of size at most s ⋅ polylog(s), which is within a polylogarithmic factor of theoretical lower bounds. The construction substantially improves previous quadratic upper bounds in quadrangles of order (s,s²), in particular in the well-studied case of the elliptic quadrics Q^-(5,s).

Related

• The known maximal partial ovoids of size q²-1 of Q(4,q)
• On elliptic ovoids and their rosettes in a classical generalized quadrangle of even order
• Ovoids of Generalized Quadrangles of Order (q, q²-q) and Delsarte Cocliques in Related Strongly Regular Graphs
• An infinite family of m-ovoids of Q(4,q)
• On the classification of low degree ovoids of Q⁺(5,q)