Update: Some new results on lower bounds on (n,r)-arcs in PG(2,q) for q≤ 31

2021/06/10 by Michael Braun, Braun, Michael
Computer Science · Mathematics · #51A05 #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2106.05908

openalex publication_date 2021/06/10 · openalex created_date 2021/06/22 · openalex updated_date 2026/07/28

Abstract

An (n,r)-arc in PG(2,q) is a set B of points in PG(2,q) such that each line in PG(2,q) contains at most r elements of B and such that there is at least one line containing exactly r elements of B. The value m_r(2,q) denotes the maximal number n of points in the projective geometry PG(2,q) for which an (n,r)-arc exists. By explicitly constructing (n,r)-arcs using prescribed automorphisms and integer linear programming we obtain some improved lower bounds for m_r(2,q): m₁₀(2,16)≥ 144, m₃(2,25)≥ 39, m₁₈(2,25)≥ 418, m₉(2,27)≥ 201, m₁₄(2,29)≥ 364, m₂₅(2,29)≥ 697, m₂₅(2,31)≥ 734. Furthermore, we show by systematically excluding possible automorphisms that putative (44,5)-arcs, (90,9)-arcs in PG(2,11), and (39,4)-arcs in PG(2,13) -- in case of their existence -- are rigid, i.e. they all would only admit the trivial automorphism group of order 1. In addition, putative (50,5)-arcs, (65,6)-arcs, (119,10)-arcs, (133,11)-arcs, and (146,12)-arcs in PG(2,13) would be rigid or would admit a unique automorphism group (up to conjugation) of order 2.

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