On sizes of complete arcs in PG(2,q)
Daniele Bartoli, Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
aa r X i v : . [ m a t h . C O ] M a y On sizes of complete arcs in
P G (2 , q ) Daniele Bartoli a , Alexander A. Davydov b , Giorgio Faina a , Stefano Marcugini a ,Fernanda Pambianco a, ∗ a Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia, Via Vanvitelli 1,Perugia, 06123, Italy b Institute for Information Transmission Problems, Russian Academy of Sciences,Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russia
Abstract
New upper bounds on the smallest size t (2 , q ) of a complete arc in the projective plane P G (2 , q ) are obtained for 853 ≤ q ≤ q ∈ T ∪ T where T = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } , T = { , , , , , , , , , , , , , } . From these new bounds it follows that for q ≤ q = 2693 , t (2 , q ) < . √ q holds. Also, for q ≤ t (2 , q ) < . √ q. It is showedthat for 23 ≤ q ≤ q ∈ T ∪ { , , } , the inequality t (2 , q ) < √ q ln . q is true. Moreover, the results obtained allow us to conjecture that this estimate holdsfor all q ≥ . The new upper bounds are obtained by finding new small complete arcswith the help of a computer search using randomized greedy algorithms. Also new con-structions of complete arcs are proposed. These constructions form families of k -arcs in P G (2 , q ) containing arcs of all sizes k in a region k min ≤ k ≤ k max where k min is of order q or q while k max has order q . The completeness of the arcs obtained by the newconstructions is proved for q ≤ ≤ q ≤ q > P G (2 , q ) arepresented for 169 ≤ q ≤
349 and q = 1013 , . Keywords:
Projective plane, Complete arcs, Small complete arcs, Spectrum ofcomplete arc sizes
1. Introduction
Let
P G (2 , q ) be the projective plane over the Galois field F q . An k -arc is a set of k points no three of which are collinear. An k -arc is called complete if it is not containedin an ( k + 1)-arc of P G (2 , q ). For an introduction in projective geometries over finitefields, see [26, 47, 49]. ∗ Corresponding author phone +39(075)5855006 fax +39(075)5855024
Email addresses: [email protected] (Daniele Bartoli), [email protected] (Alexander A.Davydov), [email protected] (Giorgio Faina), [email protected] (Stefano Marcugini), [email protected] (Fernanda Pambianco)
Preprint submitted to Elsevier October 3, 2017 n [27, 28] the close relationship between the theory of k -arcs, coding theory andmathematical statistics is presented. In particular, a complete arc in a plane P G (2 , q ) , points of which are treated as 3-dimensional q -ary columns, defines a parity check matrixof 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 [53, 54] andthey are related to optimal coverings arrays [24] and to superregular matrices [29].One of the main problems in the study of projective planes, which is also of interestin coding theory, is finding of the spectrum of possible sizes of complete arcs.A great part of this work is devoted to upper bounds on t (2 , q ), the smallest sizeof a complete arc in P G (2 , q ). Also we propose new constructions of complete arcs in P G (2 , q ) and consider the spectrum of their possible sizes.Surveys of results on the sizes of plane complete arcs, methods of their constructionand comprehension of the relating properties can be found in [3–5, 16, 25–28, 36, 42–45, 47–53]. In particular, as it is noted in [27, 28], the following idea of Segre [48] andLombardo-Radice [36] is fruitful: the points of the arc are chosen, with some exceptions,among the points of a conic or a cubic curve. We use this idea for constructions ofcomplete arcs and for finding the spectrum of arc sizes, see Sections 2, 5, and 6.The maximum size m (2 , q ) of a complete arc in P G (2 , q ) is well known. It holds that m (2 , q ) = (cid:26) q + 1 if q odd q + 2 if q even . On the other hand, finding an estimation of the minimum size t (2 , q ) is a hard openproblem.Problems connected with small complete plane arcs are considered in [3–5, 8, 9, 14,17, 19–23, 26, 30, 35, 37–39, 46–48, 50–53], see also the references therein.We denote the aggregates of q values: T = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } ; T = { , , , , , , , , , , , , , } ; T = { , , } ; Q = { , , , , } = { , , , , } ; N = { , , , , , , , , , , } . The exact values of t (2 , q ) are known only for q ≤ , see [37] and recent work [38]where the equalities t (2 ,
31) = t (2 ,
32) = 14 are proven. Also, there are the followinglower bounds (see [46, 47]): t (2 , q ) > (cid:26) √ q + 1 for any q √ q + for q = p h , p prime, h = 1 , , . Let t (2 , q ) be the smallest known size of a complete arc in P G (2 , q ). For q ≤ t (2 , q ) (up to June 2009) are collected in [5, Tab. 1] whence it follows that2 (2 , q ) < √ q for q ≤
841 . In [19], see also [23], complete (4 √ q −
4) -arcs are obtainedfor q = p odd, q ≤ q = 2401. In [9, 14] complete (4 √ q − q = 64 , , . By the results above-cited it holds that t (2 , q ) < √ q for 2 ≤ q ≤ , q ∈ Q. (1.1)For even q = 2 h , 11 ≤ h ≤
15, the smallest known sizes of complete n -arcs in P G (2 , q )are obtained in [9], see also [5, p. 35]. They are as follows: t (2 , ) = 201 , t (2 , ) =307 , t (2 , ) = 461 , t (2 , ) = 665 , t (2 , ) = 993 . Also, (6 √ q − P G (2 , q ) ,q = 4 h +1 , are constructed in [8]; for h ≤ P G (2 , ) . Let t ( P q ) be the size of the smallest complete arc in any (not necessarily Galois)projective plane P q of order q . In [30], for sufficiently large q , the following result isproved (we give it in the form of [28, Tab. 2.6]): t ( P q ) ≤ d √ q log c q, c ≤ , (1.2)where c and d are constants independent of q (i.e. universal constants). The logarithmbasis is not noted as the estimate is asymptotic.In this work, by computer search using randomized greedy algorithms (see Section 2),new small complete arcs in P G (2 , q ) are obtained for all q ∈ T ∪ T and for 853 ≤ q ≤ . For q = 601 , , the new complete arcs arose from a theoretical study on orbitsof subgroups, helped by computer [22]. From the sizes of the new arcs, with the use of(1.1) and [5, Tab. 1], [8], the following theorems result (see also Theorems 3.2, 3.3, and4.1 for more details). Theorem 1.1. In P G (2 , q ) , the following holds. t (2 , q ) < . √ q for q ≤ , q = 2693 , .t (2 , q ) < . √ q for q ≤ .t (2 , q ) < . √ q for q ∈ T . Theorem 1.2. In P G (2 , q ) ,t (2 , q ) < √ q ln . q for ≤ q ≤ , q ∈ T ∪ T . (1.3)Moreover, the study of the values of t (2 , q ) allows us to conjecture that the estimate(1.3) holds for all q ≥
23 and the last estimate of Theorem 1.1 is right for all q ≤ Conjecture 1.3. In P G (2 , q ) ,t (2 , q ) < √ q ln . q for q ≥ . (1.4) t (2 , q ) < √ q for q ≤ . Regarding the spectrum of complete arc sizes, we note (going after [28, p. 209]) thatin literature, complete arcs have constructed with sizes approximately q (see [2, 5, 13,18, 26, 33, 34, 36, 42, 44, 45, 49]), q (see [1, 31, 50, 51, 55]), q (see [31, 52]), 2 q . (see [50] where such the arcs are constructed for q > ). In particular, for even q ≥ ( q + 4)-arc [26]. Important results on the spectrum of complete arc3izes are collected in [28, Th. 2.6] where it is noted, for example, that in P G (2 , q ) with q square there exists a complete ( q − √ q + 1)-arc. In [32], large complete arcs in P G (2 , q n )are defined and new infinite families of the such arcs are constructed.Much attention is given to ( q + 5)-arcs and ( q + 7)-arcs sharing ( q + 3) pointswith a conic for q odd [2, 4, 5], [12, Rem. 3], [13, 18, 33, 34, 44]. It is proved that for allodd q there is a complete ( q + 5)-arc [34], see also [2]. Also, a complete ( q + 7)-arcexists at least for the following odd q :25 ≤ q ≤
167 [3, Tab. 2],[5, Sec. 2, Tab. 2],[16, Tab. 2.4],[34, Introduction]; q ≡ , q ≤ q ≡ , q ≤
337 [18]; (1.5) q = 2 bt − , t odd prime, b = 1 , q ≡ q ≡ q ≤
167 the known sizes of complete arcs in
P G (2 , q ) are collected in [3, Tab. 2],[5, Tab. 2], [16, Tab. 2.4].In this work new Constructions A, B, C of complete arcs are proposed, see Section 5.These constructions form families of complete k -arcs in P G (2 , q ) containing arcs of allsizes k in a region k min ≤ k ≤ k max where k min is of order q or q while k max hasorder q . The completeness of the arcs obtained by the new constructions is proved for q ≤ ≤ q ≤ q > Theorem 1.4.
Constructions A,B, and C of Section form families of complete k -arcsin P G (2 , q ) containing arcs of all sizes k in the following regions: (i) Construction A : (cid:22) q + 83 (cid:23) ≤ k ≤ q + 52 , q prime , ≤ q ≤ , ≤ q ≤ , q = 73 , , , . (ii) Construction B : (cid:22) q + 83 (cid:23) ≤ k ≤ (cid:22) q + 42 (cid:23) , q is a prime power,128 ≤ q ≤ , ≤ q ≤ , q = 89 , , , . (iii) Construction C : q + 134 ≤ k ≤ q + 52 , q ≡ is a prime power,347 ≤ q ≤ , ≤ q ≤ ,q = 199 , , , , , , , , , , . For the given q , in order to show that arcs obtained by a construction are complete weshould calculate by computer some special value, say L q (see Definitions 5.4, 5.14, 5.21)and check if L q ≤ R q where R q = ⌊ ( q − ⌋ for Constructions A, B and R q = ( q − q ≤ ≤ q ≤ L q < √ q ln q and the difference R q − L q has a tendency toincreasing when q grows, see Theorems 5.7(iii),(iv), 5.17(iii),(iv), 5.24(iii)(iv). It allowsus to conjecture the following, cf. Conjecture 5.26. Conjecture 1.5.
The assertions of Theorem hold also for all q > . k -arc of Constructions A and B contains k − k -arc of Construction C contains k − k -arcs containing k − P G (3 , q ), k -caps with k − P G ( v, n ) of growing dimensions v . Also, infinite families oflarge complete arcs in P G (2 , q n ) with growing n can be obtained by constructions of [32]using arcs of Constructions A, B, C.In this work, using Constructions A, B, C and randomized greedy algorithms, newcomplete arcs in P G (2 , q ) are obtained for 169 ≤ q ≤
349 and q = 1013 , . In Section 2 we describe the greedy algorithms used for obtaining new arcs. InSection 3 we collect the known and new upper bounds on t (2 , q ) for q ≤ q ∈ T ∪ T . The bounds are represented by tables, where values of t (2 , q ) are written,and by the corresponding relations. In Section 4 we give the upper bounds on t (2 , q ) inthe form of (1.3) and substantiate Conjecture 1.3. In Section 5 new Constructions A, B,C of complete arcs are described. Finally, in Section 6 we present new sizes of completearcs in P G (2 , q ) with 169 ≤ q ≤
349 and q = 1013 ,
2. An approach to computer search
In this paper for computer search we use the randomized greedy algorithms [3,Sec. 2],[11, Sec. 2] that are convenient for relatively large q and for obtaining examplesof different sizes of complete arcs. At every step an algorithm minimizes or maximizesan objective function f but some steps are executed in a random manner. The numberof these steps and their ordinal numbers have been taken intuitively. Also, if the sameextremum of f can be obtained in distinct ways, one way is chosen randomly.We begin to construct a complete arc by using a starting set of points S . At the i -th step one point is added to the set and we obtain a point set S i . As the value of theobjective function f we consider the number of points in P G (2 , q ) that lie on bisecantsof the set obtained. For small arcs we look for the maximum of the objective function f. For the spectrum of arc sizes we use both the maximum and the minimum of f .On every of “random” steps we take d q of randomly chosen uncovered points of P G (2 , q ) and compute the objective function f adding each of these d q points to S i . Thepoint providing the extremum is included into S i . The value of d q is given intuitivelydepending upon q, upon the number of chosen points (i.e. | S i − | ), and upon the currenttask (small arcs or the spectrum of arc sizes). For example, one can put d q = 1 forfinding of the spectrum and d q = 100 β with β = 1 , , . . . for small arcs.As S we can use a subset of points of an arc obtained in previous stages of the search.Also, for finding the spectrum of arc sizes it is fruitful to take as S a part of points of5 conic. A generator of random numbers is used for a random choice. To get arcs withdistinct sizes, starting conditions of the generator are changed for the same set S . Inthis way the algorithm works in a convenient limited region of the search space to obtainexamples improving the size of the arc from which the fixed points have been taken.In order to obtain arcs with new sizes one should make sufficiently many attemptswith the randomized greedy algorithms. For small arcs, the so called predicted sizesconsidered in Section 4 are useful for understanding if a good result have been obtained.If the result is not close to the predicted size, the attempts should be continued.Note also that arcs with sizes close to t (2 , q ) usually are obtained as a byproductwhen we execute the computer search for the smallest arcs using a few attempts.
3. Small complete k -arcs in P G (2 , q ), q ≤ , q ∈ T Throughout the paper, in all tables we denote A q = (cid:4) a q √ q − t (2 , q ) (cid:5) where a q = q ≤ , q ∈ Q . ≤ q ≤ , q = 2693 , , q / ∈ Q ≤ q ≤ , q / ∈ { , } . . Also, in all tables, B q is a superior approximation of t (2 , q ) / √ q. For q ≤ t (2 , q ) (up to June 2009) are collected in [5, Tab. 1]. Inthis work we obtained small arcs with new sizes for q ∈ T . The new arcs are obtainedby computer search, based on the randomized greedy algorithms. Complete 90-arcs for q = 601 ,
661 came from a theoretical study on orbits of subgroups, helped by computer,see [22]. These arcs are announced also in [10, Tab. 1]. A complete 104-arc for q = 709is obtained by the greedy algorithm with the starting point set taking from [22]. Thecurrent values of t (2 , q ) for q ≤
841 are given in Table 1. The data for q ∈ T improvingresults of [5, Tab. 1] are written in Table 1 in bold font. The exact values t (2 , q ) = t (2 , q )are marked by the dot “ (cid:5) ”. In particular, due to the recent result [38] we noted the values t (2 ,
31) = t (2 ,
32) = 14 . From Table 1 and the results of [19, 23], on complete (4 √ q − q = p (seeIntroduction) we obtain Theorem 3.1 improving and extending the results of [5, Th. 1]. Theorem 3.1. In P G (2 , q ) , the following holds. t (2 , q ) < √ q for ≤ q ≤ , q ∈ Q. (3.1) t (2 , q ) ≤ √ q for ≤ q ≤ , q = 101; t (2 , q ) < . √ q for ≤ q ≤ t (2 , q ) < . √ q for ≤ q ≤ , q = 521 , , , , . Also, t (2 , q ) ≤ √ q − for ≤ q ≤ , q = 23 , , , , , , , , t (2 , q ) ≤ √ q − for ≤ q ≤ , q = 317 , , , t (2 , q ) ≤ √ q − for ≤ q ≤ , q = 16 , , , , , t (2 , q ) ≤ √ q − for ≤ q ≤ , q = 421 , , , , , , able 1 The smallest known sizes t = t (2 , q ) < √ q of complete arcs in planes PG(2 , q ) , q ≤ .A q = (cid:4) √ q − t (2 , q ) (cid:5) , B q ≥ t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q q t A q B q (cid:5) .
83 128 34 11 3 .
01 347 67 7 3 .
599 94 3 3 . (cid:5) .
31 131 36 9 3 .
15 349 67 7 3 .
601 90 8 3 . (cid:5) .
00 137 37 9 3 .
17 353 68 7 3 .
607 95 3 3 . (cid:5) .
69 139 37 10 3 .
14 359 69 6 3 .
613 96 3 3 . (cid:5) .
27 149 39 9 3 .
20 361 69 7 3 .
617 96 3 3 . (cid:5) .
13 151 39 10 3 .
18 367 70 6 3 .
619 96 3 3 . (cid:5) .
00 157 40 10 3 .
373 70 7 3 .
625 96 4 3 . (cid:5) .
12 163 41 10 3 .
22 379 71 6 3 .
631 97 3 3 .
13 8 (cid:5) .
22 167 42 9 3 .
26 383 71 7 3 .
641 98 3 3 .
16 9 (cid:5) .
25 169 42 10 3 .
24 389 72 6 3 .
66 643 99 2 3 . (cid:5) .
173 43 9 3 .
397 73 6 3 .
67 647 99 2 3 . (cid:5) .
30 179 44 9 3 .
29 401 74 6 3 .
70 653 100 2 3 . (cid:5) .
181 44 9 3 .
28 409 74 6 3 .
659 100 2 3 . (cid:5) .
40 191 46 9 3 .
33 419 76 5 3 .
661 90 12 3 .
27 12 (cid:5) .
193 46 9 3 .
421 76 6 3 .
673 101 2 3 .
29 13 (cid:5) .
42 197 47 9 3 .
35 431 77 6 3 .
677 102 2 3 . (cid:5) .
52 199 47 9 3 .
34 433 77 6 3 .
683 102 2 3 . (cid:5) .
48 211 49 9 3 .
38 439 78 5 3 .
691 103 2 3 .
37 15 9 2 .
47 223 51 8 3 .
443 78 6 3 .
701 104 1 3 . .
50 227 51 9 3 .
449 79 5 3 .
73 709 104 2 3 .
43 16 10 2 .
229 51 9 3 .
38 457 80 5 3 .
719 106 1 3 . .
63 233 52 9 3 .
461 80 5 3 .
727 106 1 3 . .
58 239 53 8 3 .
463 80 6 3 .
729 104 4 3 . .
48 241 53 9 3 .
467 81 5 3 .
733 107 1 3 . .
243 53 9 3 .
479 82 5 3 .
739 107 1 3 . .
57 251 55 8 3 .
487 83 5 3 .
743 108 1 3 . .
75 256 55 9 3 .
491 83 5 3 .
751 108 1 3 . .
257 55 9 3 .
44 499 84 5 3 .
757 109 1 3 . .
62 263 56 8 3 .
46 503 85 4 3 .
79 761 109 1 3 . .
81 269 57 8 3 .
48 509 85 5 3 .
77 769 110 0 3 . .
271 57 8 3 .
512 86 4 3 .
81 773 111 0 4 . .
277 58 8 3 .
521 86 5 3 .
77 787 112 0 4 . .
97 281 59 8 3 .
52 523 86 5 3 .
77 797 112 0 3 . .
97 283 59 8 3 .
529 87 5 3 .
809 113 0 3 . .
05 289 60 8 3 .
53 541 89 4 3 .
83 811 113 0 3 . .
293 60 8 3 .
547 89 4 3 .
81 821 114 0 3 . .
06 307 62 8 3 .
54 557 90 4 3 .
82 823 114 0 3 . .
10 311 63 7 3 .
563 91 3 3 .
827 115 0 4 . .
07 313 63 7 3 .
569 91 4 3 .
829 115 0 4 . .
11 317 63 8 3 .
571 92 3 3 .
839 115 0 3 . .
10 331 65 7 3 .
577 92 4 3 .
841 112 4 3 . .
14 337 66 7 3 .
587 93 3 3 .
127 35 10 3 .
343 66 8 3 .
57 593 94 3 3 . (2 , q ) ≤ √ q − for ≤ q ≤ , q = 509 , , , , , t (2 , q ) ≤ √ q − for ≤ q ≤ , q = 569 , , , , , , , q ∈ Q ; t (2 , q ) < √ q − for ≤ q ≤ , q = 661 , , , q ∈ Q ; t (2 , q ) ≤ √ q − for ≤ q ≤ , q = 709 , , , q ∈ Q ; t (2 , q ) < √ q − for ≤ q ≤ , q = 841 , q ∈ Q. In Table 2, the current values of t (2 , q ) for 853 ≤ q ≤ q = p with t (2 , q ) = 4 √ q − Theorem 3.2. In P G (2 , q ) , the following holds. t (2 , q ) < . √ q for q ≤ , q = 2693 , . (3.2) t (2 , q ) < . √ q for q ≤ , q = 1021 , , , , , t (2 , q ) < . √ q for q ≤ , q = 1297 , , , , t (2 , q ) < . √ q for q ≤ , q = 1601 , , , , , , , t (2 , q ) < . √ q for q ≤ , q = 2011 , , , , , , , . Also, t (2 , q ) < . √ q − for q ≤ , q = 1117 , , , , t (2 , q ) < . √ q − for q ≤ , q = 1213 , , , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 1381 , , t (2 , q ) < . √ q − for q ≤ , q = 1471 , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 1601 , , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 1733 , , , , , t (2 , q ) < . √ q − for q ≤ , q = 1879 , t (2 , q ) < . √ q − for q ≤ , q = 1979 , , , , t (2 , q ) < . √ q − for q ≤ , q = 2063 , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 2203 , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 2287 , , , , , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 2417 , , , , , , . In Table 3, the current values of t (2 , q ) for 2609 ≤ q ≤ t (2 , q ) < . √ q are written in bold font.Values of t (2 , q ) for relatively great q ∈ T ∪ T are given in Table 4. The notation D q ( ) is explained in the next section. 8 able 2 The smallest known sizes t = t (2 , q ) < . √ q of complete arcs in planes P G (2 , q ) , ≤ q ≤ , A q = (cid:4) a q √ q − t (2 , q ) (cid:5) , B q ≥ t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q q t A q B q
853 117 14 4 .
01 1277 150 10 4 .
20 1693 178 7 4 .
33 2141 205 3 4 . .
04 1279 150 10 4 .
20 1697 178 7 4 .
33 2143 205 3 4 . .
03 1283 150 11 4 .
19 1699 178 7 4 .
32 2153 205 3 4 . .
02 1289 151 10 4 .
21 1709 178 8 4 .
31 2161 205 4 4 . .
02 1291 151 10 4 .
21 1721 180 6 4 .
34 2179 207 3 4 . .
05 1297 151 11 4 .
20 1723 180 6 4 .
34 2187 207 3 4 . .
04 1301 152 10 4 .
22 1733 180 7 4 .
33 2197 208 2 4 . .
03 1303 151 11 4 .
19 1741 181 6 4 .
34 2203 208 3 4 . .
06 1307 152 10 4 .
21 1747 181 7 4 .
34 2207 208 3 4 . .
05 1319 153 10 4 .
22 1753 182 6 4 .
35 2209 208 3 4 . .
06 1321 153 10 4 .
21 1759 182 6 4 .
34 2213 209 2 4 . .
07 1327 153 10 4 .
21 1777 183 6 4 .
35 2221 209 3 4 . .
06 1331 154 10 4 .
23 1783 183 7 4 .
34 2237 210 2 4 . .
08 1361 156 10 4 .
23 1787 183 7 4 .
33 2239 210 2 4 . .
07 1367 156 10 4 .
22 1789 184 6 4 .
36 2243 210 3 4 . . . .
34 2251 211 2 4 .
961 120 4 3 . .
24 1811 184 7 4 .
33 2267 211 3 4 . .
09 1381 157 10 4 .
23 1823 186 6 4 .
36 2269 212 2 4 . .
08 1399 159 9 4 .
26 1831 186 6 4 .
35 2273 212 2 4 . .
07 1409 159 9 4 .
24 1847 187 6 4 .
36 2281 213 1 4 . .
09 1423 160 9 4 .
25 1849 187 6 4 .
35 2287 213 2 4 . .
04 1427 160 9 4 .
24 1861 188 6 4 .
36 2293 213 2 4 . .
09 1429 161 9 4 .
26 1867 189 5 4 .
38 2297 213 2 4 . .
10 1433 161 9 4 .
26 1871 189 5 4 .
37 2309 214 2 4 . .
09 1439 161 9 4 .
25 1873 189 5 4 .
37 2311 214 2 4 . .
11 1447 162 9 4 .
26 1877 189 5 4 .
37 2333 215 2 4 . .
10 1451 162 9 4 .
26 1879 189 6 4 .
37 2339 216 1 4 . . .
25 1889 190 5 4 .
38 2341 216 1 4 . .
12 1459 163 8 4 .
27 1901 191 5 4 .
39 2347 216 2 4 . .
11 1471 163 9 4 .
25 1907 191 5 4 .
38 2351 216 2 4 . .
10 1481 164 9 4 .
27 1913 191 5 4 .
37 2357 217 1 4 . .
11 1483 164 9 4 .
26 1931 192 5 4 .
37 2371 217 2 4 . .
11 1487 164 9 4 .
26 1933 192 5 4 .
37 2377 216 3 4 . .
12 1489 165 8 4 .
28 1949 193 5 4 .
38 2381 217 2 4 . .
11 1493 165 8 4 .
28 1951 194 4 4 .
40 2383 218 1 4 . .
13 1499 165 9 4 .
27 1973 195 4 4 .
40 2389 218 1 4 . .
13 1511 166 8 4 .
28 1979 195 5 4 .
39 2393 218 2 4 . .
12 1523 167 8 4 .
28 1987 196 4 4 .
40 2399 219 1 4 . .
12 1531 167 9 4 .
27 1993 196 4 4 . . .
14 1543 167 9 4 .
26 1997 196 5 4 .
39 2411 220 0 4 . .
13 1549 169 8 4 .
30 1999 196 5 4 .
39 2417 220 1 4 . .
15 1553 169 8 4 .
29 2003 197 4 4 .
41 2423 220 1 4 . able 2 (continue)The smallest known sizes t = t (2 , q ) < . √ q of complete arcs in planes P G (2 , q ) , ≤ q ≤ , A q = (cid:4) a q √ q − t (2 , q ) (cid:5) , B q ≥ t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q q t A q B q .
13 1559 169 8 4 .
29 2011 197 4 4 .
40 2437 221 1 4 . .
15 1567 170 8 4 .
30 2017 197 5 4 .
39 2441 221 1 4 . .
14 1571 170 8 4 .
29 2027 198 4 4 .
40 2447 221 1 4 . .
16 1579 170 8 4 .
28 2029 198 4 4 .
40 2459 222 1 4 . .
16 1583 171 8 4 .
30 2039 199 4 4 .
41 2467 223 0 4 . .
17 1597 172 7 4 .
31 2048 199 4 4 .
40 2473 223 0 4 . .
15 1601 172 8 4 .
30 2053 200 3 4 .
42 2477 223 0 4 . .
17 1607 172 8 4 .
30 2063 200 4 4 .
41 2503 225 0 4 . .
18 1609 172 8 4 .
29 2069 200 4 4 .
40 2521 225 0 4 . .
17 1613 173 7 4 .
31 2081 201 4 4 .
41 2531 226 0 4 . .
19 1619 173 8 4 .
30 2083 201 4 4 .
41 2539 226 0 4 . .
17 1621 173 8 4 .
30 2087 201 4 4 .
40 2543 226 0 4 . .
19 1627 174 7 4 .
32 2089 202 3 4 .
42 2549 226 1 4 . .
18 1637 174 8 4 .
31 2099 202 4 4 .
41 2551 227 0 4 . .
17 1657 175 8 4 .
30 2111 203 3 4 .
42 2557 227 0 4 . .
19 1663 176 7 4 .
32 2113 203 3 4 .
42 2579 228 0 4 . .
18 1667 176 7 4 .
32 2129 204 3 4 .
43 2591 229 0 4 . .
19 1669 176 7 4 .
31 2131 204 3 4 .
42 2593 229 0 4 . . . . q ∈ Q, we use the results of [9, 19, 23], see also [5, p. 35]. In Table 4, for q ∈ T we use the results of [8, 9], see also [5, p. 35] and Introduction. The rest of sizes k for small complete k -arcs in Tables 2, 3 and 4 is obtained in this work by computersearch with the help of the randomized greedy algorithms.Note that a complete 199-arc in P G (2 , P G (2 , P G (2 , q = 2 , , , see Introduction.From Tables 3 and 4, we obtain Theorem 3.3. Theorem 3.3. In P G (2 , q ) , the following holds. t (2 , q ) < . √ q for q ≤ . (3.3) t (2 , q ) < . √ q for q ≤ , q = 3221 , , , , , , , , , t (2 , q ) < . √ q for q ≤ , q = 4099 , , , , , , , , , , , , , , , , , , . Also, t (2 , q ) < √ q − for q ≤ , q = 3413 , , , , , , , , , , , , , , , , able 3 The smallest known sizes t = t (2 , q ) < . √ q of complete arcs in planes P G (2 , q ) , ≤ q ≤ A q = (cid:4) a q √ q − t (2 , q ) (cid:5) , B q ≥ t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q q t A q B q .
51 3079 253 24 4 .
56 3571 277 21 4 .
64 4073 299 20 4 . .
52 3083 253 24 4 .
56 3581 278 21 4 .
65 4079 300 19 4 . .
52 3089 254 23 4 .
58 3583 277 22 4 .
63 4091 300 19 4 . .
51 3109 255 23 4 .
58 3593 278 21 4 .
64 4093 300 19 4 . .
51 3119 255 24 4 .
57 3607 278 22 4 .
63 4096 301 19 4 . .
53 3121 255 24 4 .
57 3613 279 21 4 .
65 4099 300 20 4 . .
52 3125 256 23 4 .
58 3617 278 22 4 .
63 4111 301 19 4 . .
52 3137 257 23 4 .
59 3623 279 21 4 .
64 4127 302 19 4 . .
51 3163 257 24 4 .
57 3631 280 21 4 .
65 4129 302 19 4 . .
53 3167 258 23 4 .
59 3637 280 21 4 .
65 4133 302 19 4 . .
52 3169 258 23 4 .
59 3643 281 20 4 .
66 4139 302 19 4 . .
52 3181 259 23 4 .
60 3659 281 21 4 .
65 4153 301 21 4 . .
52 3187 258 24 4 .
58 3671 282 20 4 .
66 4157 303 19 4 . . .
59 3673 282 21 4 .
66 4159 302 20 4 . .
53 3203 259 23 4 .
58 3677 282 21 4 .
66 4177 303 20 4 . .
52 3209 260 23 4 .
59 3691 283 20 4 .
66 4201 305 19 4 . .
52 3217 261 22 4 .
61 3697 283 21 4 .
66 4211 305 19 4 . .
54 3221 261 22 4 .
60 3701 283 21 4 .
66 4217 305 19 4 . .
53 3229 260 24 4 .
58 3709 284 20 4 .
67 4219 305 19 4 . .
52 3251 262 23 4 .
60 3719 284 20 4 .
66 4229 305 20 4 . .
54 3253 261 24 4 .
58 3721 284 21 4 .
66 4231 306 19 4 . .
53 3257 263 22 4 .
61 3727 284 21 4 .
66 4241 306 19 4 . .
53 3259 263 22 4 .
61 3733 284 21 4 .
65 4243 306 19 4 . . .
60 3739 283 22 4 .
63 4253 306 20 4 . .
53 3299 264 23 4 .
60 3761 286 20 4 .
67 4259 307 19 4 . .
52 3301 264 23 4 .
60 3767 286 20 4 .
66 4261 307 19 4 . .
55 3307 265 22 4 .
61 3769 286 20 4 .
66 4271 307 19 4 . .
55 3313 265 22 4 .
61 3779 286 21 4 .
66 4273 306 20 4 . .
54 3319 266 22 4 .
62 3793 287 20 4 .
67 4283 308 19 4 . .
54 3323 265 23 4 .
60 3797 287 21 4 .
66 4289 308 19 4 . .
54 3329 266 22 4 .
62 3803 287 21 4 .
66 4297 308 19 4 . .
55 3331 266 22 4 .
61 3821 288 21 4 .
66 4327 310 18 4 . .
54 3343 265 24 4 .
59 3823 289 20 4 .
68 4337 311 18 4 . .
55 3347 266 23 4 .
60 3833 289 20 4 .
67 4339 311 18 4 . .
55 3359 267 22 4 .
61 3847 290 20 4 .
68 4349 311 18 4 . .
54 3361 267 22 4 .
61 3851 290 20 4 .
68 4357 311 19 4 . .
56 3371 268 22 4 .
62 3853 289 21 4 .
66 4363 310 20 4 . .
55 3373 268 22 4 .
62 3863 290 20 4 .
67 4373 312 18 4 . .
55 3389 268 23 4 .
61 3877 291 20 4 .
68 4391 313 18 4 . .
55 3391 269 22 4 .
62 3881 291 20 4 .
68 4397 313 18 4 . .
53 3407 270 21 4 .
63 3889 291 20 4 .
67 4409 314 18 4 . .
56 3413 269 23 4 .
61 3907 292 20 4 .
68 4421 314 18 4 . able 3 (continue)The smallest known sizes t = t (2 , q ) < . √ q of complete arcs in planes P G (2 , q ) , ≤ q ≤ A q = (cid:4) a q √ q − t (2 , q ) (cid:5) , B q ≥ t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q q t A q B q .
55 3433 270 22 4 .
61 3911 292 20 4 .
67 4423 315 17 4 . .
57 3449 272 21 4 .
64 3917 293 19 4 .
69 4441 315 18 4 . .
56 3457 271 22 4 .
61 3919 292 21 4 .
67 4447 314 19 4 . .
53 3461 272 22 4 .
63 3923 293 20 4 .
68 4451 316 17 4 . .
54 3463 272 22 4 .
63 3929 293 20 4 .
68 4457 315 18 4 . .
57 3467 272 22 4 .
62 3931 293 20 4 .
68 4463 315 19 4 . .
57 3469 272 22 4 .
62 3943 294 19 4 .
69 4481 315 19 4 . .
56 3481 272 23 4 .
62 3947 294 20 4 .
68 4483 317 17 4 . .
57 3491 273 22 4 .
63 3967 294 20 4 .
67 4489 316 19 4 . .
57 3499 273 22 4 .
62 3989 296 19 4 .
69 4493 317 18 4 . .
57 3511 274 22 4 .
63 4001 296 20 4 .
68 4507 318 17 4 . .
57 3517 274 22 4 .
63 4003 296 20 4 .
68 4513 318 17 4 . .
58 3527 275 21 4 .
64 4007 297 19 4 .
70 4517 317 19 4 . .
57 3529 275 22 4 .
63 4013 296 20 4 .
68 4519 319 17 4 . .
57 3533 275 22 4 .
63 4019 296 20 4 .
67 4523 318 18 4 . .
58 3539 275 22 4 .
63 4021 296 21 4 .
67 4547 319 18 4 . .
57 3541 276 21 4 .
64 4027 296 21 4 .
67 4549 319 18 4 . .
57 3547 276 21 4 .
64 4049 298 20 4 .
69 4561 319 18 4 . .
58 3557 277 21 4 .
65 4051 299 19 4 . .
57 3559 276 22 4 .
63 4057 298 20 4 . , , , t (2 , q ) < √ q − for q ≤ , q = 3659 , , , , , , , , , , , , , , , , , t (2 , q ) < √ q − for q ≤ , q = 3919 , , , , , , , , , , , , , , , , , , , , , , t (2 , q ) < √ q − for q ≤ , q = 4357 , , , , , , .
4. Observations of t (2 , q ) values We look for upper estimates of the collection of t (2 , q ) values from Tables 1-4 in theform (1.2), see [30] and [28, Tab. 2.6]. For definiteness, we use the natural logarithms.Let c be a constant independent of q . We introduce D q ( c ) and D q ( c ) as follows: t (2 , q ) = D q ( c ) √ q ln c q,t (2 , q ) = D q ( c ) √ q ln c q. (4.1)12 able 4 The smallest known sizes t = t (2 , q ) of complete arcs in planes P G (2 , q ) with q ∈ T ∪ T A q = (cid:4) √ q − t (2 , q ) (cid:5) , B q > t (2 , q ) / √ qq t A q B q D q ( ) q t A q B q D q ( ) q t A q B q D q ( )4597 321 18 4 .
74 0 . .
76 0 . .
87 0 . .
74 0 . .
74 0 . .
98 0 . .
75 0 . .
78 0 .
665 5 .
20 0 . .
74 0 . .
82 0 .
993 5 .
49 0 . .
76 0 . .
84 0 . .
99 0 . .
77 0 . .
86 0 . D aver ( c, q ) be the average value of D q ( c ) calculated in the region q ≤ q ≤ q ∈ T under condition q / ∈ N. From Tables 1-4, we obtain Observation 1.
Observation 1.
Let ≤ q ≤ q ∈ T , under condition q / ∈ N . Then (i) When q grows, D q (0 . has a tendency to decreasing. ( ii) When q grows, D q (0 . has a tendency to increasing. (iii) When q grows , the values of D q (0 . oscillate about the average value D aver (0 . , . Also, . < D q (0 . < . ≤ q ≤ , . < D q (0 . < . ≤ q ≤ , . < D q (0 . < . ≤ q ≤ , . < D q (0 . < . ≤ q ≤ , . < D q (0 . < . ≤ q. (4.2)Data for relatively big q , collected in Table 4, in large confirm Observation 1.By Observation 1 it seems that the values of D q (0 .
75) and D q (0 .
75) are sufficientlyconvenient for estimates of t (2 , q ) and t (2 , q ) . From Tables 1-4, we obtain Theorem 4.1.
Theorem 4.1. In P G (2 , q ) ,t (2 , q ) < . √ q ln . q for ≤ q ≤ , q ∈ T ∪ T . (4.3)In Theorem 1.2 we slightly rounded the estimate (4.3 ).The graphs of values of √ q ln . q , √ q ln . q , t (2 , q ), and √ q ln . q are shown onFig. 2 where √ q ln . q is the top curve and √ q ln . q is the bottom one.One can see on Fig. 2 that always t (2 , q ) < √ q ln . q and, moreover, when q grows,the graphs √ q ln . q and t (2 , q ) diverge so that positive difference √ q ln . q − t (2 , q )increases.We denote b t (2 , q ) = D aver (0 . , √ q ln . q, ∆ q = t (2 , q ) − b t (2 , q ) , P q = 100∆ q t (2 , q ) % . (4.4)13
500 1000 1500 2000 2500 3000 3500 4000 4500 50000.9450.950.9550.96 q
Figure 1: The values of D q (0 .
75) for 173 ≤ q ≤ q / ∈ N . D aver (0 . , . One can treat b t (2 , q ) as a predicted value of t (2 , q ). Then ∆ q is the difference betweenthe smallest known size t (2 , q ) of complete arcs and the predicted value. Finally, P q isthis difference in percentage terms of the smallest known size. Observation 2.
Let ≤ q ≤ or q ∈ T , q / ∈ N . Then − . < ∆ q < . . (4.5) − . < P q < .
67% if 131 ≤ q ≤ , − . < P q < .
51% if 1009 ≤ q ≤ , − . < P q < .
38% if 2003 ≤ q ≤ , − . < P q < .
29% if 3001 ≤ q ≤ , − . < P q < .
23% if 4001 ≤ q. (4.6)By (4.5) and (4.6), see also Fig. 3 and 4, the upper bounds of ∆ q and P q are relativelysmall. Moreover, the upper bound of P q decreases when q grows. Therefore the valuesof ∆ q and P q are useful for computer search of small arcs.The relations (4.2)–(4.6), Theorems 1.2 and 4.1, and Figures 1–4 are the foundationfor Conjecture 1.3. 14
500 1000 1500 2000 2500 3000 3500 4000 4500 5000050100150200250300350400 q
Figure 2: The values of √ q ln . q (the top curve), √ q ln . q (the 2-nd curve), t (2 , q ) (the 3-rd curve),and √ q ln . q (the bottom curve) for 23 ≤ q ≤ Remark 4.2.
By above, √ q ln . q seems a reasonable upper bound on the currentcollection of t (2 , q ) values. It gives some reference points for computer search andfoundations for Conjecture 1.3 on the upper bound for t (2 , q ). In principle, the constant c = 0 .
75 can be sightly reduced to move the curve √ q ln c q near to the curve of t (2 , q ),see Fig. 2. For example, from Tables 1-4, Theorem 4.3 holds. Theorem 4.3. In P G (2 , q ) ,t (2 , q ) < √ q ln . q for ≤ q ≤ , q ∈ T ∪ T . (4.7)
5. Constructions of families of complete arcs in
P G (2 , q ) In the homogenous coordinates of a point ( x , x , x ) we put x ∈ { , } , x , x ∈ F q . Let F ∗ q = F q \{ } . Let ξ be a primitive element of F q . Remind that indexes of powers of ξ are calculated modulo q − . Throughout this section we use the conic C of equation x = x x . We denote pointsof C as follows: A i = (1 , i, i ) , i ∈ F q ; A d = (1 , ξ d , ξ d ) , d ∈ { , , . . . , q − } ; A ∞ = (0 , , .
500 1000 1500 2000 2500 3000 3500 4000 4500 5000−2−1.5−1−0.500.51 q
Figure 3: The values of ∆ q = t (2 , q ) − b t (2 , q ) for 173 ≤ q ≤ q / ∈ N Through this subsection, q ≥
19 is an odd prime . Let H be an integer in the region (cid:22) q − (cid:23) ≤ H ≤ q − . (5.1)We denote by V H the following ( H + 1)-subset of the conic C : V H = { A i : i = 0 , , , . . . , H } ⊂ C . (5.2)We denote the points of P G (2 , q ) : P = (0 , , , T H = (0 , , b H ) , b H = ( H + 1 if H = (cid:4) ( q − (cid:5) H if (cid:4) ( q − (cid:5) < H ≤ ( q − . (5.3)Let ℓ be the line of equation x = 0 . It is the tangent to C at A ∞ . It holds that { P , T H } ⊂ ℓ . Construction A.
Let q be an odd prime . Let H, V H , P and T H be given by (5.1)–(5.3).We construct a point ( H + 3)-set K H in the plane P G (2 , q ) as follows: K H = V H ∪ { P, T H } . The following lemma can be proved by elementary calculations.16
500 1000 1500 2000 2500 3000 3500 4000 4500 5000−1.2−1.1−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.100.10.20.30.40.50.60.7 q % Figure 4: The values of P q = 100∆ q /t (2 , q )% for 173 ≤ q ≤ q / ∈ N Lemma 5.1. (i)
Let i = j. A point (0 , , b ) is collinear with points A i , A j if and only if b = i + j. (5.4)(ii) Let i = j, a, b ∈ F q , b = a . Then a point (1 , a, b ) is collinear with A i , A j if andonly if b = a ( i + j ) − ij. (iii) Let a ∈ F q , a = i. Then a point (1 , a, i ) is collinear with P, A i . Theorem 5.2.
The ( H + 3) -set K H of Construction A is an arc in P G (2 , q ) . Proof.
By (5.1),(5.2), the sum i + j in (5.4) is running on { , , . . . , H − } where2 H − ≤ q − (cid:4) ( q − (cid:5) < H and 2 H − < ( q −
1) if H = (cid:4) ( q − (cid:5) . So, { , b H } ∩ { , , . . . , H − } = ∅ , see (5.3). Therefore P and T H do not lie on bisecantsof V H . In other side, any point of V H does not lie on the line P T H as P T H is a tangentto C in A ∞ . Theorem 5.3.
Let H be given by (5.1) . Then all points of ℓ ∪ C\V H lie on bisecantsof K H . Proof.
All points of ℓ are covered as two points P and T H of this line belong to K H . Let R and S be sets of integers modulo q, i.e. R ∪ S ⊂ F q .Let R = {− H, − ( H − , . . . , − } = { q − H, q − ( H − , . . . , q − } . By Lemma 5.1(i),points A j , A − j , P are collinear. Therefore, a point A j of C\V H with j ∈ R lies on thebisecant of K H through P and A − j where − j ∈ { H, H − , . . . , } , A − j ∈ V H . S = { b H − H, b H − ( H − , . . . , b H − , b H } . By Lemma 5.1(i), a point A j of C\V H with j ∈ S lies on the bisecant T H A b H − j where b H − j ∈ { H, H − , . . . , , } ,A b H − j ∈ V H . Let b H = 2 H +1 . Then S = { H +1 , H +2 , . . . , H +1 } . Also, by (5.3), H = (cid:4) ( q − (cid:5) whence H = ( q − v ) where v ∈ { , } and v ≡ q (mod 3). Hence, 3 H = q − v ,2 H + 1 = q − H − v + 1 ∈ { q − H, q − H − } . Let b H = 2 H. Then S = { H, H + 1 , . . . , H } . Also, by (5.3),
H > (cid:4) ( q − (cid:5) whence H > ( q − v ) where v ∈ { , } is as above. Therefore 3 H > q − v , 2 H > q − H − v ,2 H ≥ q − H − v + 1 ∈ { q − H, q − H − } . We proved that { H + 1 , H + 2 , . . . , q − } ⊆ S ∪ R . Also we showed that the points A j of C\V H with j ∈ S ∪ R are covered by bisecants of K H through P (if j ∈ R ) orthrough T H (if j ∈ S ) . In the other side,
C\V H = { A j : j = H +1 , H +2 , . . . , q − }∪{ A ∞ } where A ∞ ∈ ℓ . So, all points of C\V H are covered . Definition 5.4.
Let q be an odd prime . Let H be an integer and let P H = { P } ∪ { A i : i = 0 , , , . . . , H } . We call critical value of H and denote by H q the smallest value of H such that all pointsof the form (1 , a, b ) , a, b ∈ F q , b = a , lie on bisecants of P H . Theorem 5.5.
Let q ≥ be an odd prime . Let H q ≤ ( q − and let max { H q , (cid:22) q − (cid:23) } ≤ H ≤ q − . Then the arc K H of Construction A is complete. Proof.
We use Theorem 5.3 and Definition 5.4.In this subsection, we put q ≥
19 as we checked by computer that ( q − < H q if q ≤ . Corollary 5.6.
Let q ≥ be an odd prime . Let H q ≤ ( q − . Then Construction Aforms a family of complete k -arcs in P G (2 , q ) containing arcs of all sizes k in the region max { H q , (cid:22) q − (cid:23) } + 3 ≤ k ≤ q + 52 . If H q ≤ (cid:4) ( q − (cid:5) then cardinality of this family is equal to (cid:6) ( q + 5) (cid:7) and size of thesmallest complete arc of the family is (cid:4) ( q + 8) (cid:5) . By computer search using Lemma 5.1(ii),(iii) we obtained the following theorem.
Theorem 5.7.
Let q ≥ be an odd prime . Let H q be given by Definition . Weintroduce D q and ∆ q as follows: H q = D q √ q ln . q, ∆ q = (cid:4) ( q − (cid:5) − H q . Then the ollowing holds. (i) (cid:22) q − (cid:23) < H q ≤ q − if ≤ q ≤ and q = 79 , , , . (ii) H q ≤ (cid:22) q − (cid:23) if ≤ q ≤ , ≤ q ≤ , q = 73 , , , . (iii) H q < . √ q ln . q if ≤ q ≤ , ≤ q ≤ . (5.5)(iv) 0 ≤ ∆ q ≤ , . < D q < . , if ≤ q ≤ ≤ ∆ q ≤ , . < D q < . , if ≤ q ≤ ≤ ∆ q ≤ , . < D q < . , if ≤ q ≤ ≤ ∆ q ≤ , . < D q < . , if ≤ q ≤ . For situations (cid:4) ( q − (cid:5) < H q , the values of H q are given in Table 5. Table 5
The values H q , G q , J q for cases H q , G q > (cid:4) ( q − (cid:5) , J q > ( q − q H q G q J q q H q G q J q q H q G q J q q H q G q J q q J q
19 9 43 19 16 71 24 25 103 34 167 4223 11 47 18 18 73 27 107 36 30 179 6227 12 49 18 79 29 27 125 43 191 4929 13 53 20 19 81 29 127 39 211 5431 14 14 59 23 24 83 29 29 131 38 223 6032 15 61 22 22 89 32 139 38 343 8637 16 16 64 24 97 33 151 4241 16 19 67 24 23 101 35 163 41
Theorem 5.8.
Let q be an odd prime with ≤ q ≤ , ≤ q ≤ , or q = 73 , , , . Then Construction A forms a family of complete k -arcs in P G (2 , q ) containing arcs of all sizes k in the region (cid:22) q + 83 (cid:23) ≤ k ≤ q + 52 . Proof.
We use Corollary 5.6 and Theorem 5.7(ii).
Throughout this subsection, q ≥
32 is a prime power . Let G be an integer in theregion (cid:22) q − (cid:23) ≤ G ≤ (cid:24) q − (cid:25) . (5.6)We denote by D G the following ( G + 1)-subset of the conic C : D G = { A d : d = 0 , , , . . . , G } ⊂ C . (5.7)Clearly, A / ∈ D G . Let γ ∈ F q , γ = (cid:26) − ξ ( q − / for q odd1 for q even . (5.8)19e denote the points of P G (2 , q ) : Z = (1 , , γξ ) , B G = (1 , , γξ β G ) , β G = 2 G. (5.9)Let ℓ be the line of equation x = 0 . It is the bisecant A ∞ A of C . We have { Z, B G } ⊂ ℓ .Using (5.6),(5.8), by elementary calculations we obtained the following lemma. Lemma 5.9. (i) Let d = t. A point (0 , , b ) is collinear with points A d , A t if and only if b = ξ d + ξ t . (5.10)(ii) A point (0 , , b ) is collinear with points A d and (1 , , γU ) if and only if b = ξ d + Uξ d . (5.11) Corollary 5.10. (i)
For all q, the point P = (0 , , does not lie on any bisecant of D G . (ii) Let q be even. Then the points P, Z, A d are collinear if and only if d = 0 . Also,the points
P, B G , A d are collinear if and only if d = G . (iii) Let q ≡ . Then the points
P, Z, A d are collinear if and only if d ∈{ ( q − , ( q − } . Also, the points
P, B G , A d are collinear if and only if d ∈ { G + ( q − ,G − ( q − } . (iv) Let q ≡ . Then the points
P, Z, A d and P, B G , A d are not collinear forany d. Proof. (i) In (5.10) , the case b = 0 implies ξ d + ξ t = 0 . For even q, it is impossible . Forodd q, we obtain ξ d = − ξ t whence, by (5.8), d = t + ( q − / . By (5.6), it is impossible.(ii)-(iv) In (5.11) , the case b = 0 implies ξ d + U = 0 whence ξ d + 1 = 0 if (1 , , γU ) = Z and ξ d + ξ β G = 0 if (1 , , γU ) = B G . Remind that β G = 2 G and d ≤ q − . (ii) Here q is even but q − , , γU ) = Z, we have ξ d = 1 whence d = 0 . For (1 , , γU ) = B G , it holds that ξ d = ξ β G whence 2 d ≡ G (mod q − . So, d = G. (iii) Here both q − ( q −
1) are even. If (1 , , γU ) = Z then ξ d = − ξ ( q − / whence 2 d ≡ ( q −
1) (mod q − . It is possible if d = ( q −
1) or d = ( q − , , γU ) = B G then, by (5.8), ξ d = − ξ β G = ξ β G +( q − / whence 2 d ≡ G + ( q − / q − . So, d = G + ( q −
1) or d = G − ( q − q − ( q −
1) is odd. If (1 , , γU ) = Z then ξ d = − ξ ( q − / whence 2 d ≡ ( q −
1) ( mod q − . It is impossible. For (1 , , γU ) = B G , it holdsthat ξ d = − ξ β G = ξ G +( q − / that is impossible. Construction B.
Let q be a prime power. Assume that q . Let G, D G , Z, and B G be given by (5.6)–(5.9). We construct a point ( G + 3)-set W G in P G (2 , q ) asfollows: W G = D G ∪ { Z, B G } . From Lemma 5.1 it follows.
Lemma 5.11.
Let d = t. A point (1 , , γξ β ) is collinear with A d , A t if and only if β = d + t. (5.12)20 heorem 5.12. The ( G + 3) -set W G of Construction B is an arc. Proof.
By (5.6),(5.7), the sum d + t in (5.12) is running on { , , . . . , G − } where2 G − ≤ q − . So, { , β G } ∩ { , , . . . , G − } = ∅ , see (5.9). Therefore Z and B G donot lie on bisecants of D G . In other side, any point of D G does not lie on the line ZB G as ZB G is the bisecant of C through A ∞ and A where { A ∞ , A } ∩ D G = ∅ . Theorem 5.13.
Let q be a prime power. Assume that q . Let G begiven by (5.6) . Then all points of { P } ∪ ℓ ∪ C r D G lie on bisecants of the arc W G of Construction B. Proof.
By (5.6),(5.7), { A , A ( q − / } ⊂ D G . So, the point P is covered by Corol-lary 5.10(ii),(iii).All points of ℓ are covered as two points Z and B G of this line belong to W G . Throughout this proof, R and S are sets of integers modulo q − . It can be said that R and S are sets of indexes of powers of ξ. Let R = {− G, − ( G − , . . . , − } = { q − − G, q − − ( G − , . . . , q − } . ByLemma 5.11, points A t , A − t , Z are collinear. Therefore, a point A t of C r D G with t ∈ R lies on the bisecant of W G through A − t and Z where − t ∈ { G, G − , . . . , } , A − t ∈ D G . Let S = { β G − G, β G − ( G − , . . . , β G − , β G } . By Lemma 5.11, a point A t of C r D G with t ∈ S lies on the bisecant B G A β G − t where β G − t ∈ { G, G − , G − , . . . , , } , A β G − t ∈ D G . As β G = 2 G, we have S = { G, G + 1 , . . . , G } . Also, G ≥ (cid:4) ( q − (cid:5) . If 3 | ( q − G ≥ ( q −
1) whence 2 G ≥ q − G − . If 3 ∤ ( q −
1) then G ≥ ( q −
2) whence2 G ≥ q − G − . We proved that { G + 1 , G + 2 , . . . , q − } ⊆ S ∪ R . Also we showed that the points A t of C r D G with t ∈ S ∪ R are covered by bisecants of W G either through Z (if t ∈ R )or through B G (if t ∈ S ). In the other side, C r D G = { A t : t = G + 1 , G + 2 , . . . , q − } ∪ { A ∞ , A } where { A ∞ , A } ⊂ ℓ . So, all points of C r D G are covered . Definition 5.14.
Let q be a prime power. Let q . For integer G, let Z G = { Z } ∪ { A d : d = 0 , , , . . . , G } . We call critical value of G and denote by G q the smallest value of G such that all points(1 , a, b ) with a ∈ F ∗ q , b ∈ F q , b = a , and all points (0 , , b ) with b ∈ F ∗ q , lie on bisecantsof Z G . Theorem 5.15.
Let q ≥ be a prime power. Let q . If G q ≤ (cid:6) ( q − (cid:7) and max { G q , (cid:22) q − (cid:23) } ≤ G ≤ (cid:24) q − (cid:25) , then the arc W G of Construction B is complete. Proof.
We use Theorem 5.13 and Definition 5.14.In this subsection, we put q ≥
32 as we checked by computer that (cid:6) ( q − (cid:7) < G q if q ≤ . orollary 5.16. Let q be a prime power. If G q ≤ (cid:6) ( q − (cid:7) then Con-struction B forms a family of complete k -arcs in P G (2 , q ) containing arcs of all sizes k in the region max { G q , (cid:22) q − (cid:23) } + 3 ≤ k ≤ (cid:24) q − (cid:25) + 3 = (cid:26) ( q + 3) if q odd ( q + 4) if q even . If G q ≤ (cid:4) ( q − (cid:5) then size of the smallest complete arc of the family is (cid:4) ( q + 8) (cid:5) . By computer search using Lemmas 5.1, 5.9, 5.11 we obtained the following theorem.
Theorem 5.17.
Let q be a prime power. Let G q be given by Definition . We introduce d q and δ q as follows: G q = d q √ q ln . q, δ q = (cid:4) ( q − (cid:5) − G q . Thenthe following holds. (i) (cid:22) q − (cid:23) < G q ≤ (cid:24) q − (cid:25) if ≤ q ≤ and q = 97 , , . (ii) G q ≤ (cid:22) q − (cid:23) , if ≤ q ≤ , ≤ q ≤ , q = 89 , , , . (iii) G q < . √ q ln . q if ≤ q ≤ , ≤ q ≤ . (5.13)(iv) 1 ≤ δ q ≤ , . < d q < . , if ≤ q ≤ ≤ δ q ≤ , . < d q < . , if ≤ q ≤ ≤ δ q ≤ , . < d q < . , if ≤ q ≤ ≤ δ q ≤ , . < d q < . , if ≤ q ≤ . For situations (cid:4) ( q − (cid:5) < G q , the values of G q are given in Table 5. Theorem 5.18.
Let q be a prime power. Let ≤ q ≤ , ≤ q ≤ , or q = 89 , , , . Then Construction B forms a family of complete k -arcsin P G (2 , q ) containing arcs of all sizes k in the region (cid:22) q + 83 (cid:23) ≤ k ≤ (cid:26) ( q + 3) if q odd ( q + 4) if q even . Proof.
We use Corollary 5.16 and Theorem 5.17(ii).
Throughout this subsection, q ≥
27 is a prime power and also q ≡ . Let J be an integer in the region q − ≤ J ≤ q − . (5.14)Notations D J , B J , and β J are taken from (5.7) and (5.9) with substitution G by J. Using(5.14), it is easy to see that Corollary 5.10(i),(iv), Theorem 5.12 and their proofs holdfor D J , B J , and β J as well as for D G , B G , and β G . Construction C.
Let q ≡ prime power. . Let
P, J, D J , Z, and B J begiven by (5.3),(5.14),(5.7), and (5.9). We construct a point ( J + 4)-set E J in P G (2 , q ) asfollows: E J = D J ∪ { P, Z, B J } . heorem 5.19. The ( J + 4) -set E J of Construction C is an arc. Proof.
The set D J ∪ { Z, B J } is an arc due to Theorem 5.12. By Corollary 5.10(i),(iv),the point P does not lie on bisecants of D J and D J ∪ { Z, B J } . Finally, P, Z, B J are notcollinear. Theorem 5.20.
Let q ≡ be a prime power. Let J be given by (5.14) . Thenall points of ℓ ∪ C r D J lie on bisecants of the arc E J of Construction C. Proof.
All points of ℓ are covered as two points Z and B J of this line belong to E J . Throughout this proof, R , S , and T are sets of integers modulo q − . It can be saidthat R , S , and T are sets of indexes of powers of ξ. We act similarly to the proof ofTheorem 5.13.Let R = {− J, − ( J − , . . . , − } = { q − − J, q − − ( J − , . . ., q − } . By Lemma 5.11,a point A t of C r D J with t ∈ R lies on the bisecant of E J through A − t and Z where − t ∈ { J, J − , . . . , } , A − t ∈ D J . Let S = { β J − J, β J − ( J − , . . . , β J − , β J } . By Lemma 5.11, a point A t of C r D J with t ∈ S lies on the bisecant B J A β J − t where β J − t ∈ { J, J − , J − , . . . , , } , A β J − t ∈ D J . Let T = { ( q − , ( q −
1) + 1 , . . . , ( q −
1) + J } . By (5.8),(5.10), points
P, A t , and A t +( q − / are collinear. Therefore, a point A t of C r D J with t ∈ T lies on the bisecant P A t +( q − / where t + ( q − ∈ { q − , q, . . . , q − J } = { , , . . . , J } , A t +( q − / ∈ D J . As β J = 2 J, we have S = { J, J + 1 , . . . , J } where 2 J ≥ ( q − − , see (5.14). Also,by (5.14), ( q −
1) + J ≥ (3 q −
5) while q − − J ≤ (3 q −
5) + 1 . We proved that { J + 1 , J + 2 , . . . , q − } ⊆ S ∪ R ∪ T . Also we showed that the points A t of C r D J with t ∈ S ∪ R ∪ T are covered by bisecants of E J either through Z (if t ∈ R ) or through B J (if t ∈ S ) or, finally, through P (if t ∈ T ). In the other side, C r D J = { A t : t = J + 1 , J + 2 , . . . , q − } ∪ { A ∞ , A } where { A ∞ , A } ⊂ ℓ . So, allpoints of C r D J are covered . Definition 5.21.
Let q ≡ prime power. For integer J, let Q J = { P, Z } ∪ { A d : d = 0 , , , . . . , J } . We call critical value of J and denote by J q the smallest value of J such that all points(1 , a, b ) with a ∈ F ∗ q , b ∈ F q , b = a , and all points (0 , , b ) with b ∈ F ∗ q , lie on bisecantsof Q J . Theorem 5.22.
Let q ≥ be a prime power. Let q ≡ . If J q ≤ ( q − and max { J q , q − } ≤ J ≤ q − , then the arc E J of Construction C is complete. Proof.
We use Theorem 5.20 and Definition 5.21.In this subsection, we put q ≥
27 as we checked by computer that ( q − < J q if q ≤ . orollary 5.23. Let q ≡ be a prime power. If J q ≤ ( q − then Construc-tion C forms a family of complete k -arcs in P G (2 , q ) containing arcs of all sizes k in theregion max { J q , q − } + 4 ≤ k ≤ q + 52 . If J q ≤ ( q − then cardinality of this family is equal to ( q − and size of the smallestcomplete arc of the family is ( q + 13) . By computer search using Lemmas 5.1, 5.9, 5.11 we obtained the following theorem.
Theorem 5.24.
Let q ≡ be a prime power. Let J q be given by Defini-tion . We introduce r q and θ q as follows: J q = r q √ q ln . q, θ q = ( q − − J q . Then it holds that (i) q − < J q ≤ q − if ≤ q ≤ and q = 211 , , J q ≤ q − , if ≤ q ≤ , ≤ q ≤ ,q = 199 , , , , , , , , , , J q < . √ q ln . q if ≤ q ≤ , ≤ q ≤ ≤ θ q ≤ , . < r q < . , if ≤ q ≤ ≤ θ q ≤ , . < r q < . , if ≤ q ≤ ≤ θ q ≤ , . < r q < . , if ≤ q ≤ ≤ θ q ≤ , . < r q < . , if ≤ q ≤ . For situations ( q − < J q , the values of J q are given in Table 5. Theorem 5.25.
Let q ≡ be a prime power. Let ≤ q ≤ or q =199 , , , , , , , , , , . Then Construction C forms a familyof complete k -arcs in P G (2 , q ) containing arcs of all sizes k in the region q + 134 ≤ k ≤ q + 52 . Proof.
We use Corollary 5.23 and Theorem 5.24(ii).Basing on Theorems 5.7, 5.17, 5.24 and taking into account that √ q ln . q < √ q ln q , √ q ln . q < √ q ln q , we conjecture the following, cf. Conjecture 1.5. Conjecture 5.26.
Let H q , G q , J q be given by Definitions , , . Let for H q , q be prime while for G q and J q it holds that q is a prime power. Finally, let q for G q and q ≡ for J q . Then the following holds. H q ≤ (cid:22) q − (cid:23) if q ≥ G q ≤ (cid:22) q − (cid:23) if q ≥ J q ≤ q − if q ≥ .H q < √ q ln q if q ≥ G q < √ q ln q if q ≥ J q < √ q ln q if q ≥ . (5.16)24 emark 5.27. It is interesting to compare the relations (5.5),(5.13),(5.15),(5.16) withTheorems 1.2, 4.1, 4.3 and Conjecture 1.3 and to compare also computer results providingTheorems 5.7, 5.17, 5.24 with Tables 1 and 2. One can see that the upper estimates of t (2 , q ) , H q , G q , and J q have the same structure and the values of t (2 , q ) , H q , G q , and J q have a close order. This seems to be natural as almost all points of P G (2 , q ) lie onbisecants of P H q , Z G q , and Q J q , see Definitions 5.4, 5.14 and 5.21. Remark 5.28.
The complete arcs of Constructions A, B, C can be used as startingobjects in inductive constructions. For example, for even q , arcs of Construction Bcan be used in constructions of [9, Ths 1.1,3.14-3.17,4.6-4.8]. In that way, one cangenerate infinite sets of families of complete caps in projective spaces P G ( v, n ) of growingdimensions v . For every v , constructions of [9] can obtain a complete cap from everycomplete arc of Construction B. Also, it can be shown that in Constructions A, B, C allpoints not on conic are external . So, the arcs of Constructions A and C for q ≡ P G (2 , q n ) with growing n can be obtained.
6. On the spectrum of possible sizes of complete arcs in
P G (2 , q ) The main known results on the spectrum of possible sizes of complete arcs in
P G (2 , q )are given in Introduction with the corresponding references. Taking into account theresults cited in Introduction, we denote M q = ( q + 4) for even q ( q + 7) for odd q included to (1.5) ( q + 5) for odd q not included to (1.5 ) . We suppose that the smallest known sizes t (2 , q ) are given in Tables 1-4 of this paper. Theorem 6.1. In P G (2 , q ) with ≤ q ≤ , and q = 1013 , , there are complete k -arcs of all the sizes in the region t (2 , q ) ≤ k ≤ M q . Proof.
For 25 ≤ q ≤
167 the assertion of the theorem follows from [3, Tab. 2] and[5, Tab. 2]. For 169 ≤ q ≤
349 and q = 1013 , t (2 , q ) in the end of Section 2. Note also that therest of sizes for all 169 ≤ q ≤
349 and q = 1013 , S . For this weused consequently subsets of cardinality approximately 15% , , ,
30% of the coniccardinality q + 1. Conjecture 6.2.
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