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Tables, Bounds and Graphics of Sizes of Complete Arcs in the Plane PG(2, Q) for All Q 321007 and Sporadic Q in [323761

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Tables, Bounds and Graphics of Sizes of Complete Arcs in the Plane PG(2, Q) for All Q 321007 and Sporadic Q in [323761 Tables, bounds and graphics of sizes of complete arcs in the plane PG(2; q) for all q 321007 and sporadic q in [323761 ::: 430007] obtained≤ by an algorithm with fixed order of points (FOP)∗ Daniele Bartoli Dipartimento di Matematica e Informatica, Universit`adegli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: [email protected] Alexander A. Davydov Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences Bol'shoi Karetnyi per. 19, Moscow, 127051, Russian Federation. E-mail: [email protected] Giorgio Faina Dipartimento di Matematica e Informatica, Universit`adegli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: [email protected] Alexey A. Kreshchuk Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences Bol'shoi Karetnyi per. 19, Moscow, 127051, Russian Federation. E-mail: [email protected] Stefano Marcugini and Fernanda Pambianco arXiv:1404.0469v3 [math.CO] 3 Jan 2018 Dipartimento di Matematica e Informatica, Universit`adegli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: stefano.marcugini,fernanda.pambianco @unipg.it f g ∗The research of D. Bartoli, G. Faina, S. Marcugini, and F. Pambianco was supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project \Geometrie di Galois e strutture di inci- denza") and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM). The research of A.A. Davydov and A.A. Kreshchuk was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (project 14-50-00150). This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC Kurchatov Institute, http://ckp.nrcki.ru/. 1 Abstract. In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane PG(2; q) is proposed to construct small complete arcs. In each step, the algorithm adds to a current arc the first point in the fixed order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate that PG(2; q) contains a sufficient number of relatively small complete arcs. Also, in the previous papers, it is shown that the type of order on the points of PG(2; q) is not relevant. A complete lexiarc in PG(2; q) is a complete arc obtained by the algorithm FOP using the lexicographical order of points. In this work, we collect and analyze the sizes of complete lexiarcs in the following regions: all q 321007, q prime power; 15 sporadic q's in the≤ interval [323761 ::: 430007], see (1.10). In the work [9], the smallest known sizes of complete arcs in PG(2; q) are collected for all q 160001, q prime power. The sizes of complete arcs, collected in this work and in [9], ≤ provide the following upper bounds on the smallest size t2(2; q) of a complete arc in the projective plane PG(2; q): p p t2(2; q) < 0:998 3q ln q < 1:729 q ln q for 7 q 160001; p p ≤ ≤ t2(2; q) < 1:05 3q ln q < 1:819 q ln q for 7 q 321007: ≤ ≤ Our investigations and results allow to conjecture that the bound t2(2; q) < 1:05p3q ln q < 1:819pq ln q holds for all q 7. It is noted that sizes of the random complete arcs and≥ complete lexiarcs behave similarly. This work can be considered as a continuation and development of the paper [11]. Mathematics Subject Classification (2010). 51E21, 51E22, 94B05. Keywords. Projective planes, complete arcs, small complete arcs, upper bounds, algo- rithm FOP, randomized greedy algorithms 1 Introduction. The main results Let PG(2; q) be the projective plane over the Galois field Fq of q elements. An n-arc is a set of n points no three of which are collinear. An n-arc is called complete if it is not contained in an (n + 1)-arc of PG(2; q). For an introduction to projective geometries over finite fields see [56,58,59,80]. The relationship among the theory of n-arcs, coding theory, and mathematical statistics is presented in [40,58,59,63,66,67]. In particular, a complete arc in the plane PG(2; q); the points of which are treated as 3-dimensional q-ary columns, defines a parity check matrix of a q-ary linear code with codimension 3, Hamming distance 4, and covering radius 2. Arcs can be interpreted as linear maximum distance separable (MDS) codes [85, Sec. 7], [87] and 2 they are related to optimal coverings arrays [54], superregular matrices [61], and quantum codes [26]. A point set S PG(2; q) is 1-saturating if any point of PG(2; q) S is collinear with two points in S [19{21,⊂ 24, 37, 38, 49, 73, 88]. In the literature, 1-saturatingn sets are also called \saturated sets", \spanning sets", and \dense sets". The points of a 1-saturating set in PG(2; q) form a parity check matrix of a linear covering code with codimension 3, Hamming distance 3 or 4, and covering radius 2. An open problem is to find small 1-saturating sets (respectively, short covering codes). A complete arc in PG(2; q) is, in particular, a 1-saturating set; often the smallest known complete arc is the smallest known 1-saturating set [24,37,38,49,75]. Let `1(2; q) be the smallest size of a 1-saturating set in PG(2; q). In [21], combining approaches of [73] and [8], the following upper bound is proved for arbitrary q: p r q `1(2; q) q(3 ln q + ln ln q) + + 3: (1.1) ≤ 3 ln q Let t2(2; q) be the smallest size of a complete arc in the projective plane PG(2; q). One of the main problems in the study of projective planes, which is also of interest in coding theory, is the determination of the spectrum of possible sizes of complete arcs. In particular, the value of t2(2; q) is interesting. Finding estimates of the minimum size t2(2; q) is a hard open problem, see e.g. [60, Sec. 4.10]. This paper is devoted to upper bounds on t2(2; q). Surveys of results on the sizes of plane complete arcs, methods of their construction, and the comprehension of the relating properties can be found in [5, 6, 8{11, 13, 15, 18, 23, 33, 34, 43,49,55,56,58{60,62,76,77,80{82,84{86]. Problems connected with small complete arcs in PG(2; q) are considered in [1{18,22,24, 25,27,31{38,41{53,55{60,62,64,68{71,74{77,79{86,89{92], see also the references therein. The exact values of t2(2; q) are known only for q 32; see [2,24,31,32,42,52,56,57,70,71]. The following lower bounds hold (see [3,27,79] and≤ the references therein): p2q + 1 for any q; t2(2; q) > 1 h (1.2) p3q + 2 for q = p ; p prime, h = 1; 2; 3: Let t( q) be the size of the smallest complete arc in any (not necessarily Desarguesian) P projective plane q of order q. In [62], for q large enough, the following result is proved by probabilistic methodsP : C t( q) pq ln q; C 300; (1.3) P ≤ ≤ where C is a constant independent of q (so-called universal or absolute constant). Surveys and results of random constructions for geometrical objects in PG(2; q) can be found in [19{21,28,45,62,65,73]. In PG(2; q), complete arcs are obtained by algebraic constructions (see [59, p. 209]) with 1 1 0:9 10 sizes approximately 3 q [1, 13, 64, 81{83, 92], 4 q [13, 64, 84], 2q for q > 7 [81], and 3 2:46q0:75 ln q for big prime q [53]. It is noted in [45, Sec. 8] that the smallest size of a complete arc in PG(2; q) obtained via algebraic constructions is cq3=4 (1.4) where c is a universal constant [84, Sec. 3], [85, Th. 6.8]. There is a substantial gap between the known upper bounds and the lower bounds on t2(2; q), see (1.2){(1.4). The gap is reduced if one considers the lower bound (1.2) for complete arcs and the upper bound (1.1) for 1-saturating sets. However, though complete arcs are 1-saturating sets, they represent a narrower class of objects. Therefore, for complete arcs, one may not use the bound (1.1) directly. Nevertheless, the common nature of complete arcs and 1-saturating sets allows to hope for upper bounds on t2(2; q) similar to (1.1). The hope is supported by numerous experimental data and some probabilistic conjectures, see below. In [62, p. 313], it is noted (with reference to the work [27]) that in a preliminary report of 1989, J.C. Fisher obtained by computer search complete arcs in many planes of small orders and conjectured that the average size of a complete arc in PG(2; q) is about p3q log q. In [8], see also [7], an attempt to obtain a theoretical upper bound on t2(2; q) with the main term of the form cpq ln q, where c is a small universal constant, is done.
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