Planar arcs

Abstract

Let p denote the characteristic of F q , the finite field with q elements. We prove that if q is odd then an arc of size q + 2 − t in the projective plane over F q , which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most t + p ⌊log p t⌋ , provided a certain technical condition on t is satisfied.This implies that if q is odd then an arc of size at least q − √ q + √ q/p + 3 is contained in a conic if q is square and an arc of size at least q − √ q + 7 2 is contained in a conic if q is prime. This is of particular interest in the case that q is an odd square, since then there are examples of arcs, not contained in a conic, of size q − √ q + 1, and it has long been conjectured that if q = 9 is an odd square then any larger arc is contained in a conic.These bounds improve on previously known bounds when q is an odd square and for primes less than 1783. The previously known bounds, obtained by Segre [23], Hirschfeld and Korchmáros [15] [16], and Voloch [29] [30], rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse-Weil theorem and the Stöhr-Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents.Arcs in the projective plane over F q of size q and q + 1, q odd, were classified by Segre [22] in 1955. In this article, we complete the classification of arcs of size q − 1 and q − 2.

Arcs in finite projective spaces

“…Corollary 57 proves a conjecture from Hirschfeld, Korchmáros and Torres [26,Remark 13.63]. It was proven in [5] as a corollary of Theorem 54.…”

“…In contrast to the previous bounds the following theorem does not rely on Hasse-Weil or Stöhr-Voloch. It is proved using the tensor of a planar arc Theorem 44 from [5]. It illustrates that in this case the tensor form (of degree t in each component) is stronger than the algebraic hypersurface (which is of degree 2t).…”

“…Proof. Let C k denote the Reed-Solomon code obtained from the normal rational curve in PG(k − 1, q) as in (5).…”

“…It is stronger in that one does not require m = 2 in the case of q odd, but weaker in the sense that the multi-homogeneous form F does not necessarily come from a form φ(Z) on PG(k − 1, q) under substitution. This theorem is from [5] and it is proved using the scaled coordinate-free lemma of tangents (Lemma 29).…”

“…Using the projection theorem this leads to a proof of the MDS conjecture for k ≤ √ q − √ q/p + 2. The following two corollaries are from [5].…”