Arcs in \mathbb F_q²

2020/03/07 by Roche-Newton, Oliver, Warren, Audie
#52C10 #94B27 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.03656

Abstract

An arc is a subset of \mathbb F_q² which does not contain any collinear triples. Let A(q,k) denote the number of arcs in \mathbb F_q² with cardinality k. This paper is primarily concerned with estimating the size of A(q,k) when k is relatively large, namely k=q^t for some t>0. Trivial estimates tell us that q \choose k ≤ A(q,k) ≤ q² \choose k. We show that the behaviour of A(q,k) changes significantly close to t=1/2. Below this threshold an elementary argument is used to prove that the trivial upper bound above cannot be improved significantly. On the other hand, for t ≥ 1/2+δ, we use the theory of hypergraph containers to get an improved upper bound A(q,k) ≤ q^2-t+2δ \choose k. This technique is also used to give an upper bound for the size of the largest arc in a random subset of \mathbb F_q² which holds with high probability. For example, we prove that a p-random subset Q ⊂ \mathbb F_q² with q^-3/2

Related

• Counting arcs in \mathbb F_q²
• The number of arcs in \mathbbF_q² of a given cardinality
• Tables of sizes of random complete arcs in the plane PG(2,q)
• Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane PG(2,q) for all q≤160001 and sporadic q in the interval [160801… 430007]
• Upper bounds on the smallest size of a complete arc in the plane PG(2,q)