Ti*ril lownal of Pure Science 19 (1) 2011 ISSN: 1813 - 1662 A Non PGL (3,q) k-arcs in the projective plane of order 37 Nada Yass€n Kasm Yahya Depafimeht of Mathe atics , College of Mucation , Univetsity of Mosul , Mosul , Iruq ( Received: 17 I 12013 Accepte* 1 l3 12013) I -- Abstract a set ofk points such that every line in the plane intersects it in at most two points and there is a line intersects it in exactly two points. The k-arc is complete ifthere is no k+l -arc containing it. The main purpose of this paper is to study and find the projectively distinct k- arcs, k=4,5,6,7 in PG(2,37) A k-arc in the plane PG(2,q) is though the classification and construction of the projectively distinct k-arcs and finding the group of of each projectively distinct k-arc and describing it. Also it was found that PG(2,37) has no projectivities maxlmum arc . Introduction A k-arc in projective plane PG (2,q) is a set of k A Projectively S : q, -+ o is a bijection given by a non-singular matrix T such that ifP (x'): P(x) S , then tx= xT where x'and x are coordinate vectors for P(x')and P (x),and t e Ko. points such that every llne in the plane intersects it in at most two points and there is a line intersects it in exactly two points. The k-ara is complete if there is no k+1-arc containing it. The purpose of this thesis is to classify and construct the projectivety distinct k-arcs in PG(2,37) over the Galois field GF(37). Also it was shown that PG(2,37) has no maximum k-arc. Hirschfeld [7] and Sadeh [0] showed the classification and construction ofkarcs over the Galois field GF(q), with qsl1. Younis [2]first studied classification of k-arcs in the projectivety plane PG (2,16). The classification ofthe complete k-arcs in PG (2,q) ,for q:23,25,27 has been given by Coolsaet and Sticker [5.].161 . Chao ,Kaneta [4] classification ofthe complete k-arcs in PG(2,q) ,for 23:qS29. AL-Taee [1] classified the projective distinct k-arcs for k=5,6 in the projective plane PG(2,32). (l-2-2) Fundamental theorem of geometary l7l If {pr,pr,...,p.+r},{ pr*,p:*,...,p*:*}are two sets ofpoints of PG(n,q) such that no n+l points chosen (1) from the same set lie in a prime( a prime in PG(2,q) is a line) , then there exists a unique projectivity T such that pi*: piT ,for all i in N.1r={ 1,2,...,n+2}. (2) Let S = PG(n,q), and f,: S JS be a collineation, then I:oT, where o is an automorphism and T is a projectivity. This means that if K= CF (ph),and P(Y)=P(X),; then there exists m in Nh, qj in GF(phXor (ii)in Nh*x Nh* and t in GF (ph)\{6} such that tY=X' T, where r: p- and X'=(xo',xr', . . ,x") and T=(Q;ijin N6*,whereNl*={0,1,2,...,h}and Nr,:{ 1,2 ,...,h). (l-2-3) Cyclic projectivities [7] A projectivity T which permutes the points of PG (n,q) which is denoted byO (n) in a single cycle is called a cyclic projectivity , where O (n)=(q'+t. Also this paper represents an extra step in this side , which include first study for construction and classification k-arcs in the projective plane PG (2,37). The subject matter in this paper needs long time for running computer programs which are used to find projectively distinct k-arcs in the projectiye plane of order 37 and finding groups for these k-arcs for values k=5,6. l/(q-l). For example ,the projectivity represented by the matrix: 010'l 0 0 rl (l-2) Delinition(Companion Matrix) [71 Let (x)= x' ? n-r X"-r .,, ao be any monic I o r] polynomial ,then its Companion matrix ,C(f),is given bythenxnmat x: r o0 [o l0 0 r 0 c(fF ..... on xl _a2 x2 alxl _a To l -dc(D- | I ,a l-secant is called unisecant and a 2-secant is called a*,. a bisecant. (1-2-t Theorm [1ll o'l L€t t(p) be the number of unisecants through P, where P is a point ofthe k-arc K. Let T1 be the total number ofi-secants ofK in the plane ,then: ,,1 2-Tr=k(k-ly2,Tr=kt, and To:q(q-l)/2+(t-l)/2. (l-3) Group of projectiveties o , o o rl Lo" * is (l-2-4) Delinitiotrs I7,81 A line I ofPG(2,q) is an i-secant of a (k,n)-arc K if ,llnk:i ,A 0-secant is called an external line of k-arc In particular case ,when n:3 then (x): cyclic projectivity which PG (2 ,5). 0 ---- PG(2 ,5) is a given by right multiplication on the points of o I lo o o o G" q o, 4 projective l-(p)=q+2-k= (1-2-1) Delinition 135 Tikit lournal of Pure ISSN: 1813 - 1662 Science 19 (1) 2014 (r-3-r) Detinitioo l3l [,ct V be a vector space over the field k. The general linear group GL (v) is the group of all linear automorphisms of V. The subgoup consisting of those linear automorphisms with determinant I is In this paper, the consfiuction and classification of the k-arcs which 4SkS7 has been obtained and also the group of projectvities of the projectively distinct called the special linear group SL(V). Note : when V=V(n ,q),we use the notations GL(n ,q) and SL(n ,q). Let (x)= .r -3x' - x - 2 be an irreducible monic polynomial over GF(37) then the (l-3-2) Theorem [31 The order ofGL(V) and SL(V) GL(n ,q) 1q,-qn'r (2-l) The projective platre PG (237) 3 companion matrix T of(x) are given by: rvr.fI(e, 11qlq.r)...(q",) -q(, sl-(n,q)l= q'(''rY'?.lJ(q' k-arcs are found. 0 l 0l I 0 T 0l 2 1 3] r) is a cyclic projectivity on PG(2,3 7) Let po be the point Uo=(l,0,0) then -l). ,i:0,...,1406,are the 1407 points p,=poT, of PG(2,37).(see Table(l)) (r-3-3) Delinition l3l Tab :rl Points of PG The projective general linear group PGL(V),and the projective special linear group PSL(V),are defined as follows: li lr lr z PGL(V)= GL(Vyz(GL(V)),where Z(GL(V)) is the centre ofGL(V). PSL(V): SL(vy z(SL(v). When V=V(n, q) it is customary to w te the projective groupsjust defined PGL(n ,q)and PSL(n,q). lp, l-o T, r T-l It lr t I ... I ... r 2 l.r lz zo zt r------r----------i 1405 1406 as 237t6 2 437 (l-3-4) Theorem: l3l Let Lr b€ th€ line at infinity (z:0) which contains the points:- (l-3-5) Delinition: lll The projectiv€ group denoted byPGL(n,q) is the group of all projective transformations in projective 0,1,9,29,t 52,1 56,t 82,t93,262,300,323,401,404,419,4 2s,489,s39,60 5,62 1,67 9,689,7 I 4, 7 s4,8s t,923,9 4 5,972,1034,1 195,1229,r23 1,t248,126 2,130E,1321,1353,1360and 1365 then L,=L1T1'1 , i=1,...,1407,are the lines ofPG(2,37), the 1407 lines Li are given by the rows in (Table(2)). The set PGL(n ,q) forms a group which is called a projective group. space PG(n-l,q). and Classilication of k-arcs in PG(2J7) over the Galois lield GF(37) 2. Cotrstruction Tab Lines of PG Points for each line Lines l,inel 0, I ,9,29,1 52,156,182,193,262,193,262 )ffi)23,4Ot ,404At9,425,489,539,605,621 12,l8,1262,1308,1321,619,689,714,754551,923,945872,103,1,1195,1229,1231,, t35J,1J60,1365 Line2 Line1407 l ,,I 0J0,1 53,157,183 ,r91,263)Or )21,402,405,,r20,,r26,,190,s.r0 ,606,622,640,690,1 15,755,852,924, 9,16,973,1035,1 196,1230,1232,12,19,1 263,1 309,1 322,135.1,1361, 1366 14060,818,151,155,1 81,192,261 J99 )22,&0,403,4r 8,424,,t88,538,604, ,520,578,688,7 r 3,753 ,A50,922944,971,1033,1194,1228,t230,1247 ,1261 ,1307 ,1320 1352,t359,1364 (3-l)Algorithm work used for the constructior 3.Construction k-arcs for 4SIC!7 Let Po,Pt,Pz,Pt are p0 p, (0, l, 0), p, (0, 0, l), p, = 0, 0, 0), = = = Q, l, D and classilication which ,x- (n n n rn lr 0,, tr, 2, 3 form 4-arcs points respect to fundamental theorem of projective geometry(l-3) then 4-arc is the equivalent the \ J 2-Find the points which don't lie in ,l't,Ilt .lt;.ltlr construct arcs from k=5 into ^r" respect to 4-arc which includes points k=7 ll't,Pr,l,:,lttru according to these in the following algorithm work used computer program of k-rrcs which 4slc!7 in PG(2,37). l-Determine lines which are 2- secants for arc lines 2-secants. 3-Adding these point odd into k-arc and getting 5afg . 4- Finding projectively distinct k-arcs. 5- Finding groups [9] which are fixing projectively distinct k-arcs. . 136 Tikril Joarnal of Pute Science 19 (l) 2014 6-Repeating the steps from ISSN: 1813 - 1662 2 into 5 to finding arcs,6-arcs andT-arcs. 7-Finding complete arcs for all steps ifthey exist Consider the set k={0,1,2,1109} of four points of PG(2,37)no three of them are collinear. These four points form 4-arcs,to construct Projectiyely distinct 5arcs though the Projectively distinct 4-arcs respect to definition (3-2-2) we mention four points over plaoe lines and simulation points lines which are bisecanls from points of plane PG(2,37) then add these points odd into references foul points and getting s-arcs , 5- . (3-2)Projectively Distinct s-arcs (3-2-1) The Construction of the Projectively Distinct s-arcs (3-2-2)Delirition I2l L€t k be a k-arc. the points of Pc(2,q)\k which don't belong to any bisecant of k,will be called the points ofindex zero of k . Tabl LINES xl x2 xl x4 x5 x6 x7 x8 x9 xl0 xll xl2 xl3 xl4 x15 xt6 xt7 xlE xl9 Ilt t,; I I I I I I 2 I I I x20 0 x2l 0 0 0 0 I I I I x24 I I I I I I I 2 2 ) 109 I t I I I I I I I109 27 C, 2 r 109 110!) 3r I t I109 I109 I109 I109 43 44 Ta 4 lo ll 12 Tal-- tY3 To -faTlvl lo ll 12 Y5 lY6 lY? 0 lo ll lo ll I lY8 lo lr o lr Yl0 Ylt Yl2 c, c, 35 C Ct C, 50 55 It09 94 213 C1 lr09 251 I I109 292 294 C, I109 I I l0!) t09 5 Il r It09 I Io c, l4 l6 l7 I TiTNr;STP,Tp'TP. v2 CA 1l te o ll o ll 2 Also there exist 12 arcs that have a group of respect to I I l2 It 1227 1227 1227 1227 1227 1227 1221 1227 1227 1227 1221 1227 1227 1221 1227 1227 1227 1221 1227 1227 1227 1227 170 t0 t70 l0 l0 l0 170 170 170 170 170 t70 170 170 170 170 170 170 170 170 170 170 110 110 170 170 tr-o l0 I to l0 t0 l0 l0 l0 l0 l0 l0 l0 l0 10 l0 l0 l0 l0 have the projective plane PG(2,37) consisting of369l arcs fiom projectivily distant 6-arcs and there exist 3375 arcs type of identity group I, also there exist 12 arcs that have a group of type Sa as the following table below shows. ectivG Distinct Garcs IP. P. ll(D ls ll(D l6 It(D I10,!) I 14 tq- I 109 35 2 |ll(D 2 ll(D lrs 2llloe143 , I 599 C, iacomplete 24-arcs. (&&t) The Construction of the Projcctively Distinct 6arcs To construct all 6-arcs through projectively distinct 5arcs,use the same way of constructing 5-arcs.So we IYI I 1t 09 l 109 I l0!) I 109 I All the projectively distinct s-arcs xi(i=1,2,...,24), (&3) Projectivcly Distirct Gsrcs l0 7 0 0 0 0 0 0 0 0 0 t0 170 E r 2 170 I109 I109 I 0 alrnl I10,!) 0 2 1221 I C, I10,1) I I T2 3 4 5 6 ll09 I I Tl G I 0 0 0 0 To lt: I t lGl Itt l,t it Distinct 5-arcs Pro ectiv 0 0 x23 Projectively distinct s-arcs and classif, [9] as the show in table below . 0 0 x22 by using the computer program we have 24 ll(D ll(D 1t09 type 43 55 292 C2XC2 as the following table below shows. All r" Tr, T-r, "Pl 350 lr94 ll98 lrs r4Tt rs I res I007 rr II l194 I194 7-1 I rssTis 76A 215 865 tl94 It94 198 I I194 198 l3l 506 1339 1005 t5 l9E l5 l5 tt94 r9E I194 I194 t9E t5 t5 t98 tt94 trs 198 l5 the projectively distinct 6-arcs Y,(i=1,2,---,12\, are incomplete l2-arcs 137 l5 l5 . Tiktil Journal of Purc Science I9 (1) 2011 ISSN; Table (s Pro tLrNEs ZI t& lp, & I I 0 0 0 0 72 73 ZA 2 Distinct 6-arcs PI PS P6 TO T, I109 5 5 6 6 1172 629 It94 198 ll94 ll94 t98 866 1223 l194 198 198 E l2l4 ll94 198 lr(D I , I 2 I 109 7 ll(D II(D (D 75 76 0 I 0 t z7 0 7A 0 I I I Z9 0 I 2 210 Lo 1 zl2 0 r86 l194 35 I 194 I l0!) 44 4E5 It09 55 I175 I 194 I 2), are incomp lete I 2-arcs. ztt Lr Al I the proj ectively distinct 6-arcs Zi(i= I ,2, .. ., I 109 I 109 Also there exists one group of type C7xC3 ,one group of type C2XCa,one group of type D15 and exists one group of type D5, one group oftype Da,two groups Ts Pro LINES PI P2 P3 P4 P5 FI 0 0 0 0 0 0 0 0 0 F2 F3 F4 F5 F6 F7 F8 F9 I 2 1 I I I I I I I l109 , I 10,!) 2 I109 I l0!) I t09 I 91 6 50 3 l0q (D 2 5 44 50 ll(B I t0,!) All the projectively distinct 6-arcs F1(i=l,2,...,9), are incomplete g-arcs . Also it is found 34 arcs that have Ta lt ,rl l4 1227 27 1343 ll(D ) I 0 I109 t ll94 198 tt94 t98 ll94 198 35 94 198 198 198 198 T, t5 l5 l5 l5 l5 l5 l5 l5 l5 l5 l5 l5 of type CrXCrxC, , and two groups of type CrXCI the following table below shows . as Distincl G8rcs To Tr l194 198 637 tl94 r98 l5 3t1 D,. I194 l5 G E20 373 D5 tl94 198 198 l0,l D. C,XC,XC, t194 t98 tl94 198 198 198 198 411 837 cr(c,xc, tt94 shows. Distinct Gorcs Pro P2 P3 P. P5 P6 HI It2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t I I 2 2 I 109 3 3 t<t ll94 l9E l5 I109 It94 r9E I 109 7 l09l 198 I I I 1 It09 l2 ll94 tt94 tt94 tl94 t5 l5 l5 l5 l5 l5 l5 l5 l5 l5 H7 H8 H9 Hl0 Hu Ht2 Ht3 Ht4 H15 H16 0 Ht7 It l8 Hl9 0 0 H20 0 H2l 0 0 H22 0 0 I I I I I I I I I I I I I I I I t t t 2 t t I 109 4 t09 4 t73 I 109 4 4 4 4 4 4 7 7 1 I109 I109 I 109 ll(x) , , I109 I109 I l(x) 1t09 I109 ,' I I 7 It09 2 , , ,, t r4E 40 34 I t5 t5 l5 t5 lr94 l5 c-xc. t5 2lt c*xc. It94 a group of type Cr 8s the following table below ,130 Pr I13 T, t5 c,xc. c.xc. P6 LINES H4 H5 H6 l8I3 - 1662 t09 t09 llu) I 109 I 109 138 To Tr r98 198 198 198 195 1194 s29 1194 198 r 194 198 198 198 198 291 ll94 tt94 tt94 ll94 tt94 l9E t5 42t I194 198 15 15 15 6ll 701 1122 t 323 14 198 7 470 1194 r98 7 7 1 984 lt94 lt94 198 198 198 198 198 8 1045 1203 466 483 E s50 E 15 15 l5 l5 tt94 l5 lt94 l5 lt94 t98 15 1194 Tikit Joutnal of Pure Science 19 H23 H24 0 0 , I109 l 109 l,t l4 775 253 I109 I 109 t6 t1 393 2 , r 109 27 944 I109 3I 562 tl91 , I 109 t t09 35 r063 228 I194 2 r 109 1 I t09 H27 H2E 0 I I I 0 I 0 0 0 0 0 0 I H30 H31 H32 H33 H34 , I I I I 1 I 109 , I r 109 it is found 4 arcs that P2 PT P4 P3 198 l5 r9E 15 198 198 198 l5 l5 l5 l5 r98 tt94 ll94 tt94 tlg4 tl94 Distinct 6-arcs Pro ecti Ta LINES 937 ll94 ll94 ll94 lt94 198 15 198 15 213 899 198 l5 232 655 198 l5 3 308 198 15 50 I152 198 l5 have a group of type C5 as the following table below shows. 55 distinct 6-arcs H,(i=1,2,...,34), are incomplete 34-arcs . Also tt94 , 0 0 H29 ISSN: 1813 - 1662 I I H26 }l25 All the projectively (I) 2011 P5 ,t3 P6 506 772 T, TO T, ll94 t9E l5 2 I109 16 J, ll94 t98 l5 J 0 I 2 I l0!) 241 ll94 198 l5 t I 109 27 J. 0 576 ll94 t98 l5 All the projectively distinct 6-arcs l,(i=l,2,3,4), are a group of qpe C2 as the following table b€low J, It09 I I 0 0 incomplete 4-arcs . Also it is found 245 arcs that have Te ecti PI P2 P3 P4 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 wll wl2 wl3 wl4 wl5 wl6 wl7 wlE wl9 w20 w21 w22 w23 w24 w25 w26 w21 w28 w29 w30 w3l w32 w33 shows. P5 Distillct Garcs P6 26 0 I 21109 3 0121109360 0 I 2 (x) 3 6l 0 I 21109 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I t I I t I I I I r I I I l I I I 3107 21109 3 t28 211(D 3 r.() 2 2 tt09 3 t92 tt09 3 242 21109 3 382 2ll(x) 3 387 21109 3 392 21109 3l()0 2 09 3 414 2n09 3 439 21109 3 457 2ll(x) 3 551 21109 3 554 21109 3 626 21109 3 64t 2 09 3 675 2 09 3 730 21109 3737 2n(x) 3 A)t 21109 3 934 211(x) 3 95t I 1 1 r 2llo9 31003 r 21109 3lolt r 2n09 3t024 r 2rl09 310s6 r 21109 31063 I 21109 3I06E I 21109 310,69 I 21109 3t266 139 To I 194 Tr Tr 198 tt94 r98 l5 l5 I194 198 198 198 198 t5 19E 19E 15 15 15 15 lt94 I194 1194 1194 1194 ttg4 ll94 t 194 tl94 tl94 ll94 ll94 198 198 198 l5 l5 l5 l5 l9E t5 19E 15 198 l5 19t 15 I194 198 tl94 tl94 ll94 tl94 ll94 l5 l9E t5 19E 19E l5 19t 1t9,l 198 198 1194 I9E tt94 tt94 198 1194 1194 1194 1194 1194 1194 l9E 198 19E r9E 198 15 15 15 15 15 l5 l5 l5 l5 l5 l5 19E 15 ll94 198 198 l5 tt94 19E l5 15 Tikit tounol oj Parc Science w34 w35 w36 w37 w38 w39 w40 w4l w42 w43 w44 w45 w46 w47 w48 w49 w50 w5t w52 w53 w54 w55 w56 w57 w58 w59 w60 w61 w62 w63 w64 w65 w66 w67 w68 w69 w70 w71 w72 w13 w14 w75 w76 w77 w78 w79 w80 w8r w82 w83 w84 w8s w86 w87 w88 w89 w90 ISSN: 1813 - 1662 19 (1) 2011 0 I 0 l 0 I o | 0 I 0 0 o 0 0 0 0 0 0 I l t I I I I I I 0 I 0 I 0 I 0 t 0 I 0 I 0 t 0 I 0 l 0 I 0 I 0 I 21109 31303 21109 4lsE 21109 4165 21109 4197 21109 4213 21109 4214 21109 4 406 21109 4 475 2 09 4 559 2tto9 4 632 21109 4 774 21109 4 777 21109 4 802 21109 4 853 21109 4 E65 21109 4 477 21109 4 900 21t09 4 912 21109 4 939 21109 4 980 21109 41015 21109 410E7 2 09 4 3l 21109 41136 21109 4 S7 21109 4 86 o | 21t09 41236 o I 21109 412a6 0 I 21t09 0 I 21t09 413t8 41319 0 I 21109 413E1 o I 2llo9 41397 0 I 21109 41405 0 I 2 09 5 0 0 o 0 0 0 o 0 0 0 0 t I I I l I I I r I l 64 21t09 5 202 21t09 5 205 2lto9 5 224 21109 5 238 21109 5 25E 21109 5 260 2 ttog 5 217 21109 s 2E9 21109 5 291 21109 5 3E9 21t09 s 403 0 I 21109 5 436 0 I 21109 5 437 0 I 21109 5 445 0 I 21109 5 46t 0 I 21109 5 534 0 I 21109 5 586 0 I 2ll(x) 5 759 0 I 2u(x) 5 E6l 0 I 21109 5 917 0 1 21109 51t25 0 r 2rr09 0 I 51059 21109 512t7 140 r 194 I194 I I l9,l t9,l 1194 tl94 I 194 I 194 I 194 ll91 ll94 I 194 tt94 I 194 ll91 ll94 ll94 198 198 198 198 t5 l5 l5 l5 l5 l5 t5 l9E l5 198 198 19E 198 198 198 198 198 198 198 r98 t5 l5 l5 l5 l5 r5 l5 l5 t98 t5 t5 t5 I194 198 198 t5 t 194 t98 ! 194 I 194 l9E I 194 I t94 tl94 ll91 ll91 ll94 I194 I194 tt94 I l9{ I194 ll94 ll94 198 198 198 198 198 t98 r98 198 198 198 198 198 198 I194 l9E r98 l9,l 198 198 198 198 ll94 t I 194 I t94 I 194 I 19{ ll94 I t94 I 194 I 194 I r94 tt94 l9E I r94 198 198 I t94 ll94 tt94 ll91 l5 t5 l5 l5 l5 t5 t5 l5 t5 l5 l5 t5 l5 t5 l5 r5 l9,l tl94 t5 15 I I194 I194 l5 l5 l5 r98 r98 198 198 198 198 198 198 lt94 l5 l9E 198 198 l5 ls l5 t5 l5 l5 t5 l5 l5 l5 l5 l5 r98 r98 r5 t98 t5 l5 Tikit Joumal of Pure Science w91 w92 w93 w94 w95 w96 w97 w98 w99 wl00 wl01 wl02 wt03 wl04 wt05 wt06 wl07 wl08 w109 wll0 wlll wl12 w113 wl14 wl15 w 6 w117 wl18 wl19 w120 w12l wt22 wl23 w124 w125 wl26 wt27 wl28 wl29 wl30 wl3l w132 wl33 wl34 wl35 w136 wl37 wl38 wl39 wl40 wt4l wt42 wl43 wl44 w145 wl46 wl47 19 (1) 2011 ISSN: 1813 - 1662 0 I 21109 51381 0 t 2tt09 622 0 t 21109 6 3l o I 21109 636 0 t 2 tt09 676 0 1 21109 6 83 0I 0t 21109 6 95 21109 6 212 0 I 21109 6 249 0 t 2 ttog 6 289 0 r 21109 6 309 0 1 21109 6 3rr o I 2 tto9 6 326 o t 21109 6 472 0 1 21109 6 48s 0 I 21109 6 506 0 I 21109 6 508 0 I 21109 6 525 0 1 2 09 6 585 0 1 2t109 6 643 0 1 2 09 6 685 0 1 21109 6 693 0 I 21109 6 746 0 I 21109 6 76E 0 r 21109 6 836 0 1 21109 6 885 0 I 21109 6 984 0 I 21109 61252 0 1 2tl(x) 6130s 0 1 21109 61398 ot2tto972l 0 t 2 tt09 7144 0 I 2tl09 7 2tl 0l2ll09 7 254 0 I 21109 7 299 o I 2ttw 7 344 0 t 2 tlw 7 4tl 0121109 7 471 0 t 2 tto9 7 4a6 0 I 21109 7 590 0 t 2 tt09 7 656 0 t 2ll09 0 I 7 773 2ll09 7 779 2ttw 7 479 2ll09 7 937 21109 7 958 21109 71013 21109 71060 21109 71090 21109 71092 21109 71131 21109 71161 o I 0 I 0 I 0 I 0 I 0 l 0 I 0 I 0 I 0 I zlt09 71215 0 1 21109 71223 0 I 21109 7 Da4 0 I 21109 71324 0l2tt09 842 141 ttg4 r98 15 1194 1194 198 198 198 15 15 ll94 tt94 1194 1194 lt94 1194 I194 94 1194 I I94 tt94 ttg4 1194 1194 lt94 tt94 tt94 ttg4 1194 1194 1194 tt94 tt94 ttg4 tt94 ttg4 tl94 1194 1194 1194 1194 r98 l5 198 198 198 198 198 198 198 198 198 15 15 r98 r98 r98 r98 l5 198 t5 r98 15 15 198 198 198 l5 l9E t5 198 198 15 15 15 15 15 l9E 19E 198 198 198 198 l9E ll94 r98 tt94 tt94 tt94 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 ttg4 1194 1194 ttg4 1194 1194 1194 tt94 1194 tlg4 1194 1194 1194 1194 15 t5 198 1194 l5 l5 l5 l5 l9E It94 tt94 l5 15 15 15 15 15 198 tlg4 1194 t5 15 l5 15 15 15 15 15 l5 l5 l5 15 15 t5 15 15 15 l5 15 15 15 15 15 l5 15 15 15 l5 ISSN: 1813 - 1662 Tikrit Journsl of Pure Science 19 (1) 2011 0 I 21109 8113 o I 21fi9 4244 0 I 21109 8 265 o I 21rc9 4267 0 I 21109 8 362 wl48 wl49 wt50 wtSl wl52 wl53 wl54 wl55 l 21109 E 56E l 21t09 E E44 1 2 rr09 81090 1 21109 8 4l I 21109 E lt53 I 21109 811E2 1 21109 11 99 t 21109 tt 233 I 2 t109 lt 299 I 2ll(x) ll 369 0 r 21109 38s 0 0 0 0 0 0 0 0 0 0 wr56 wl57 wt58 wl59 wr60 wl6l wl62 wt63 wl64 wl65 0 0 0 0 wr66 wt67 o I 0r wl73 wt74 wt75 wl76 wl77 wl78 wl79 wl80 0 0 0 0 0 182 wl83 wr t4 wl85 wl86 wl9l wl92 w!93 wl94 wt95 wt96 wl97 wr98 wl99 t w200 wrot w202 w203 w20,t ll 2 09 tl 21109 635 685 748 822 I I r t I 21109 l4 299 21109 l4 ss9 21109 14 885 21109 t4 9s4 21109 14 1105 21t09 14 t240 21109 t4 t2s6 0 I 21109 14 t346 0 I 2tto9 141357 0 I 21109 t4 1378 0 I 2 ll09 t4 t395 0 I 21109 16 52 wl8l wl88 wl89 wl90 2t109 lt 0 I 2 09 14 6t wlTl wl72 wlET 21109 1l 0 I 21109 lt lt70 0 I 2lto9 tt 1294 0 t 2t109 14 35 wl68 wl69 wt70 w I I I I 0 I 0 0 0 0 0 0 0 0 0 0 1 1 I 21109 t6 2t3 0 I 21109 16 355 0 I 21109 16 388 2lto9 16 436 21109 16 444 2 1109 16 st6 ttog t6 706 2lto9 16 741 I l lsl I r rsl I r rsa I r tql r98 r98 15 15 l5 lt94 198 198 198 198 198 l 194 198 tl94 198 198 198 15 15 I l9,t I 194 t 194 I194 I 194 l9E tl94 lt94 198 198 198 198 198 198 t 194 I194 I l9,l t 194 tt94 r98 I194 I194 198 198 198 198 198 198 198 t 194 ll94 I 194 I r94 tl94 ll94 ll91 I 194 I t94 I t94 ll94 tlg4 tl94 t94 I194 I194 I l191 I I 0 I 0 I 142 l5 15 15 15 15 l5 l5 15 l5 l5 l5 l5 l5 198 21109 16 t242 21109 l7 506 21109 l7 910 2 1109 l7 1042 2 1109 l7 l0E3 2 1109 t7 tt43 2 tt09 27 l5l 2 tt09 27 20t 2 tt09 27 326 2 tt09 27 366 2 lt09 27 424 2 tlo9 27 686 I I l5 15 15 15 15 15 15 15 15 I194 ll94 tls4 ll94 1 l5 15 15 15 15 t5 2 1109 16 t38 I t l5 198 I I 0 0 0 0 0 0 198 198 198 198 15 198 2 I 198 r98 t5 I 194 I 194 I 194 I t94 I I I r98 198 15 198 198 l9E l5 l5 l5 198 198 15 15 198 t5 r98 15 15 1194 1194 198 198 198 198 15 15 tl94 tlg4 lt94 ll94 r9E l5 198 198 198 l5 t t94 198 I l9il tl91 198 198 I194 r98 15 l 194 198 t5 l5 15 15 l5 l5 l5 Tikrit Journal of Purc Science 19 (1) 2011 w205 w206 w207 w208 w209 0 0 0 0 0 w2l0 0 w2 0 w2t2 w2l3 w2l4 w2l5 w216 w2l7 w218 I I I I I 0 0 1 0 I 0 0 0 0 0 0 0 I 0 I 0 I 0 1 w227 w228 0 0 I I I w229 0 1 0 0 I I I 0 1 0 I w230 0 w23l 0 w232 w233 w234 w235 w236 w237 w238 w239 w240 0 0 0 0 0 0 0 0 0 0 0 w24t w242 w243 w244 w245 All the projectively distinct are incomplete 245-arcs 2 ll09 55 278 2 I t09 55 453 2 I109 55 471 I w222 w223 w224 w225 w226 2 r r09 55 590 2 r r09 55 10.17 21109 1 I t I I ,, ,| ) I , 1 2 .' 1 55 t092 2 ll09 55 t 099 2 ll09 55 t lo,t 2 ll09 55 1259 2 tt09 94 87 2 tl09 94 396 1 I 35 1220 2 r r09 44 803 2 1109 441392 2 tlog 50lt21 2 ll09 55 22E I I w22l 0 2 I t09 I I I I w2t9 w220 2 tl09 3l 282 2 lt09 31 535 2 n09 31 565 2 r r09 3l 734 2 lt09 3t 751 2 1109 31 768 2 ll09 31 1098 2 ll09 35 E33 1 t I ISSN: 1813 - 1662 I 2 ll09 2r3 533 2 1109 213 650 2 tt09 2131274 2 tt09 232 443 2 tt09 232 407 2 tl09 251 1392 2 tl09 292 569 2 tl09 292 725 2 tt09 292 tl36 2 1109 599 5t 599 t43 s99 266 599 872 599 E77 599 l rEl 599 1212 599 134.1 I109 I109 I109 r 109 I109 I109 I t09 6-arcs W,(i=I,2,...,245), (3-4) Projectively Distitrct 7-arcs: Projectively t5 t5 l9,l r98 r98 I r 194 198 lt94 r9E t94 I194 I194 198 I lt94 It9,t 1194 tt94 1194 tl94 lt94 lt94 lt91 lt94 I194 I t94 ll91 lt94 ll9,l tt91 ll94 ttg4 tt94 l5 l5 l5 l5 l9E t5 198 198 198 198 198 198 198 198 198 198 198 198 198 198 15 15 15 r98 t5 15 15 l5 15 l5 t5 15 l5 l5 l5 l5 198 15 t98 t5 t5 t5 198 198 198 198 198 15 1194 r9E 1194 1194 1194 t98 l5 l5 t5 l5 198 t5 l9E l5 l5 l5 I t94 1194 ll94 I 194 I194 I 194 I 194 1194 tt94 (4-l)Delinitions in PG(2,37) has (292786) arcs and it is big number which is found by computer program ,we cannot classify it because there are many .rcs and it takes more than (150) hours on computer and it is not yet stopping .therefore we used more than 6 computers to run it and get results of complete arcs but the computers continue working in the same method , for this reasons we stop working on this step. Existence l5 r98 198 198 198 198 198 198 198 t5 t5 l5 t5 l5 by geometric method. Distioct 7-arcs The number of projectively distinct 7-arcs 198 In previos sections we constructed all the projectively distinct k-aras for F4,5,6,7,and we see that,k-arcs with 4SkJ are incomplete. these results are obtained by computer research. We shall now prove that in theorem(4-5-l) that complete (k,3)-arcs do not exist . (3-4-l) The Cotrstruction of the lt91 lt94 tl91 [7,E1 is a set of k points ,such that there is some n but no (n+l)are collinear where n22. l-A (k,n)-arc K 2-A (k ,n)-arc is complete if,there is no (k+l,n)-arc containing it. 3- the maximum value in which (k ,n)-arc exists in the projective plane PG(2,q) is denoted of Complete (kJ)-arcs in PG(2J7) (4-2) Lemma 143 by m (n)," . Tikit Joumal oJ ISSN: I8I3 - 1662 Pure Science l9 (I) 2011 For a (k ,n)-arc K ,the following equation holds :17,81 k-l k-l (0) _, 0 denoted the number of points ofPG l- ZTi= q'+q +l Suppose(I- 2- Zttt-n(q+t) (2,37) of type (37-3m+(k+2i- l ), 2m-(k+2i- l ), m-i) , i-0,1,...,m. According to equ8tion (l) and (2) of lemma ( 4.3 ) we i=0 3- Ei(i-t)Ti= 4n-t) have, md, +(m -l)a.-r+....+a, 4- tnr=q+l 5- 2(t -t)Rt = n-l 6- ES, = q+l 7 where 13 is the total number of 3-s€cants in PG(2,37) , with 3sl:!37. Sincc rnl0,for l€3, we obtain 8-IRr =,ri tS, =(q+l-lrr, a (+3) L€mmr For a (k ,nlarc K ,the following equation brR, =i1....... T bt=L" """ t 13<k(k-l)/6 On the other hand t*mm, . .. .. ""' [71 iff (1407-37y6>228.This S"=1 1or in PG(2,q)\I( (+5) Cotrclusiotr "1; is impossible, therefore, a in PG(2,37) for complete (37,3>arc does not exist O 3*<37. The Results ($l) Projectively Distrtrce k-rrcs Tabl{l0)represens the classification The maximum value m(3)47 for which (k3)-arcs do€s not exist. (zl-$1 )Theorem of projectively distance k-arcs for k=5,6 which is N1 represents the number ofprojectively distinct arcs ,G, In PG(2,37), a complete (k,3)-arc does not exist for 3*<37. representing groups k-arcs and describtion then all Proof the projectively distinct k-arcs are incomplete arcs. this is taken from a computer work to get these resuhs, (2so)computer hours. For3:ft=37. the equalions (4) and (5) of lemma ( 4.2 ) become: Rl+R2+R3:38 R2+2R3=k-l lrt m = [(k-l) / 2 ], paItof(k-l)/2. T where l0 ix-5 lc, lc, Clr IG : TYPE OF POINT RI Ri Rr (DJ) 37-3tttr(l(-l ) 37-3trtl(k+l ) 2m{L+1) n-l (m-i) 37-3m+(k+2i.l) 2m{L+2i-l) 2m{kl) P Kd I Nr=z [(k-l) / 2 ] is the integral So the maximum value of R3 can accrue is m. Assume that r;= (k-l-2i)l , i=O,t,...,m . It is clear that m is positive for l€3 .The possible types of(k,3)arc;31X37 are given in the following in the table (n) K is complete complete (3,3)-arc does not exist in PG(2,37). For k:37, we obtain from equations (l) and (2) 13 < 222 $d t3 > 22E which is impossibte , so a complete (37.3Farc does not exist in PG(2,37). Finally ,if for 3<n<37the (n,3!arcs is complete ,then we have from equation (l) r3<n(n-l/6<(37'36y6 422 , also from equation (2)we have r3>( 1407-ny6> K is a complete (k .n)-arc ,then: (q+l-n) r;q't+l-k,with equality below .(l) the (lq3)-arc r/(t 4o7 -k)/6........(2) Now, for k:3 we obtain from the equations (l) and (2)r3<l and r3>234,which is impossible .So a holds: 16,4 I ,,t-q'+q+l-1.. If if for35ks37 , then according to lemma (4.4), we have 6r321407-k or nrS, =(4 +l-iY, (4-4) /3 Furthermore, Since m < (k-l) / 3. Sincc m S (k-l) / 2, then we have Where the summation in the equation (E) is taken over all p e K, and is taken over all Q€PG(2,qlK in the equation (9). I of(k,3!arc m(d. + dr-t +...+ dt) = z (f,i- a, )...(r+),h bigger than;ro *1, -t]l,.._t+...+at =>:4to',. Therefore,.l)'_o1 a)= nk > (Z:ako1)=3r!. This implies that mk > 3r3 or, 13 < mk - ZlSi= n 9- = 3rr...(*), = L- t-- Nr=3591 c,Ttr t2 L! 3375 10 Sr l2 I I C,r C, 12 Ic,'c, -fr It c, D.. Tr T c,' T Tn I D.--1-' --fl l*34+'- tr -Tc. C! 144 Distrncc k-arcr 3,t T4 Tikit Joumil of Pare tt Science 19 (1) 2011 ISSN: The maximum value m(3)2.37 does not exist for 3SkS37. 215 C (5-2) The Maximum Yalue m(n),, in the proj€ctive IEIi for which - 1662 (k,3)-arcs plane PG(2,q) Reference lllAL-Taee ,R.S.D,(2011),' A Non PGL(3,q) and Non PfL(3,q) k-arcs in PG(2,32) and Lower bound of 5blocking set", M.Sc, Thesis, University ofMosul . [7] Hirschfeld, J.W.P., (1979), " hojective Geometics ot'er Finile Fields", ()xford University Press, Oxford. [8] Hirschfeld, J. W. P. and Storme, L., (2001), "The packing problem in statistics,coding theory and finite projective spaces : Update 2001", Submitted. [9] Thomas, A.D. and Wood, G.V., (1980), 'Grozp farles", Shiva Publishing Ltd. ll0lSadeh, A.R.,( t984),'The classification of k-arcs and cubic surfaces with twenty seven lines over the field ofeleven elements", M.Sc. Thesis, University of [2] Azlz, 5.M., (200t), "On Lower Bound for Complete (k,n)-arc in PG(2,q)", M.Sc. Thesis, University of Mosul . [3] Biggs,N.L.,and Wlite,A.T.,( I 97 9\ " Permutation Groups and Combinalotial Sttucturcs",Ca'],]tridge University press, Camridge. [4] Chao, J.M. and Kaneta. H., (2001), "Classial arcs in PG(r, q) for 23:q:?9"Discrete Mathematics Sussex,UK. I l] Yasin, A.L., (1986), "Cubic arcs in the Eojective plane oforder eight", Ph.D. Thesis, University of Sussex . [12] Younis, H.,(1989), "Classification of k-arcs in 226,p.p.377-385. [5] Coolsaet, K. and Sticker , H.,(2008) "A full classification of complete k-arcs in PG(2,23) and PC(2.25)". Joumal of Combinatorial Designs. [6] Coolsaet, K. and Sticker, H., (2009),"4 tull classifi cation of complete k-arcs inPG(2,27)".Ghent University ,Belgium. 37 4r3.Jl the projectivety plane PG(2,16)", M.Sc. gl slli-yl 6i,,,..i ,,j li-l iili.Jl k-.r"lrfil LL. ,;r1 ;nE 4P'4 di ,JFt, (2013 ll J-/t , J-At t*t+ / 7 :jjglt 9;u Thesis, University of Mosul. trVL) ?,i l17:,.lo.rl g;u , 1--jxt LE . ---- 2Ol3 ll ) (Aldl .Li 'i.J &lr,ij dp + Y t- &li; *jLJl e! k rJS cl +.+ lEnl d, irr: rA PG(2,q) 91lJl .,.i k- , rriil k .J"!drl r1rj,! a-UrJA c6.!l lrA u.s.-L!l u,:J"Ja,Jj.J k+l-o,rg lJoJ ols.yli,jS C t:1 au rL k-*rg lldJ.L:'. !L , rk k- u,Ji ,.El ,rjt !, ,.J"!61,.JA ,r.i,J +r".ii ti-i il PG(2,37) +6-)l ,,iFJl Gk:4,5,6,7, L,.LtLl liEJt "h,j. ..r.!.1 u"ri a+j u4PG(2,37) tt) lit,r '. g! .t,i. .lJ 1,.Lg-1 dr_l:!ij, {..LiJ 145