An upper bound for the minimum weight of the dual codes of desarguesian planes

AbstractWe show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529–538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order pm where p is a prime, and m≥1. This gives words of weight 2pm+1−pm−1p−1 in the dual of the p-ary code of the desarguesian plane of order pm, and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of André planes.We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmáros and Mazzocca [Gábor Korchmáros, Francesco Mazzocca, On (q+t)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Philos. Soc. 108 (1990) 445–459]

On KM-arcs in small Desarguesian planes

In this paper we study the existence problem for KM-arcs in small Desarguesian planes. We establish a full classification of KMq,t-arcs for q <= 32, up to projective equivalence. We also construct a KM64,4-arc; as t=4 was the only value for which the existence of a KM64,t-arc was unknown, this fully settles the existence problem for q <= 64

The Kakeya problem : a gap in the spectrum and classification of the smallest examples

Kakeya sets in the affine plane are point sets that are the union of lines, one through every point on the line at infinity. The finite field Kakeya problem asks for the size of the smallest Kakeya sets and the classification of these Kakeya sets. In this article we present a new example of a small Kakeya set and we give the classification of the smallest Kakeya sets up to weight q(q + 2)/2, q/4 , both in case q even

Codes of Desarguesian projective planes of even order, projective triads and (q+t,t)-arcs of type (0,2,t)

AbstractWe study the binary dual codes associated with Desarguesian projective planes PG(2,q), with q=2h, and their links with (q+t,t)-arcs of type (0,2,t), by considering the elements of Fq as binary h-tuples. Using a correspondence between (q+t,t)-arcs of type (0,2,t) and projective triads in PG(2,q), q even, we present an alternative proof of the classification result on projective triads. We construct a new infinite family of (q+t,t)-arcs of type (0,2,t) with t=q4, using a particular form of the primitive polynomial of the field Fq

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in PG(2, q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type q/4. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type q/4

Elation KM-arcs

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 = 2(h), h > 3, an infinite family of elation KM-arcs of type q/8, and for q=2(h), where 4, 6, 7 | h an infinite family of KM-arcs of type q/16. Both families contain new examples of KM-arcs