@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Classification and Construction of (k,3)-Arcs on Projective Plane Over Galois Field GF(9) Adil M. Ahmad Amaal SH. Al-Mukhtar Fatima. F. Kareem Dept. of Mathematics/College of Education for Pure Science (Ibn AL-Haitham) University of Baghdad Received in : 27 May 2001 , Accepted in : 23 April 2001 Abstract In this work, we construct and classify the projectively distinct (k,3)-arcs in PG(2,9), where k ≥ 5, and prove that the complete (k,3)-arcs do not exist, where 5 ≤ k ≤ 13. We found that the maximum complete (k,3)-arc in PG(2,q) is the (16,3)-arc and the minimum complete (k,3)-arc in PG(2,q) is the (14,3)-arc. Moreover, we found the complete (k,3)-arcs between them. Keywords: arcs, secant, Projective plane ,Galois Field 266 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Introduction A (k,n)-arc K [1] in PG(2,q) is a set of k points, such that some n, but no n + 1 of them are collinear. A (k,3)-arc is a set of k points no four of them are collinear. A (k,n)-arc K is complete if it is not contained in a (k + 1,n)-arc. A line of the plane containing exactly i points of a (k,n)-arc is called an i-secant of K. For a (k,3)-arc each line of PG(2,q) is a 3-secant, 2-secant, 1 secant, or 0-secant. A 3-secant is called a trisecant. A point N not on a (k,n)-arc K has index i if there are exactly i trisecants of K through N. Let ci be the set of points N of index i and let Ci =  ci is the number of points N of index i. The (k,n)-arc K is complete if every point of PG(2,q) lies on some trisecant of K. Thus K is complete iff C0 = 0 [2,3]. Theorem 1: [2] Let ri be the total number of the i-secants of a (k,n)-arc K in PG(2,q), then the following equations are hold: n 2 i i 0 n i i 1 n i i 2 r q q 1 ir k(q 1) i(i 1)r k(k 1) = = = = + + = + − = − ∑ ∑ ∑ Notation 2: [2] Let ri be the total number of i-secants of a (k,n)-arc K in PG(2,q), then the type of a (k,n)-arc K with respect to its lines is denoted by (rn,rn – 1,…,r0). Definition 3: [2] Let K1 is of type (rn,rn – 1,…,r0) and K2 is of type (tn,tn – 1,…,t0), then K1 and K2 have the same type iff ri = ti for all i, and when K1 and K2 have the same type then they are projectively equivalent. Definition 4: [2]: Let Q1 and Q2 be two points in PG(2,q) which are not in arc K and let K1 = K ∪{Q1}, K 2 = K∪{Q2}, then Q1 and Q2 lie in the same set iff K1 and K2 are projectively equivalent under type of lines. The Projective Plane PG(2,9): [2] PG(2,9) contains 91 points, 91 lines, 10 points on every line and 10 lines through every point. Let Pi and Li, i = 1, 2, …, 91, be the points and the lines of PG(2,9), resp. Let i stands for the point Pi and all the points and lines of PG(2,9) are given in the table. The Classification of (5,3)-Arcs in PG(2,9): Let A ={1,2,11,21} be a set of the reference points in PG(2,9), no three of them are collinear. The distinct (5,3)-arcs can be constructed by adding to A each time one point from the remaining 87 points of PG(2,9). There are only two projectively distinct (5,3)-arcs which are: B1 = {1,2,11,21,3} and B2 = {1,2,11,21,4} 267 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 The group G(B1) has eight projectivities which has five elements of order two and two elements of order four, hence G(B1) ≅ D4. The group G(B2) ≅ D6. The Classification of (6,3)-Arcs: There are 72 points of index zero for B1. Then G(B1) partitions 72 points into 13 orbits. So we have 13(6,3)-arcs to be constructed by adding one point from each of the 13 orbits to B1. We have 79 points of index zero for B2. G(B2) partitions these points into 12 orbits. So we have 12 (6,3)-arcs to be constructed by adding one point from each of the 12 orbits to B2. So there exist twenty five (6,3)-arcs to be constructed and by projectively equivalent arcs, we have four projectivity distinct (6,3)-arcs, these arcs are: C1 = {1,2,11,21,3,12}, C2 = {1,2,11,21,3,13}, C3 = {1,2,11,21,3,32}, C4 = {1,2,11,21,4,32}. The group G(C1) ≅ S4, the group G(C2) ≅ S3, G(C3) is isomorphic to the identity group and G(C4) ≅ Z2. The Classification of (7,3)-Arcs: There groups G(Ci), i = 1, 2, 3, 4 partition the points of index zero for Ci into 133 orbits. So there exist 133 (7,3)-arcs to be constructed and by projectively equivalent arcs, we have only six (7,3)-arcs which are projectively distinct. These are: D1 = {1,2,11,21,3,12,20}, D2 = {1,2,11,21,3,12,23}, D3 = {1,2,11,21,3,12,33}, D4 = {1,2,11,21,3,13,32}, D5 = {1,2,11,21,3,32,40}, D6 = {1,2,11,21,4,32,42}. Each one of them is incomplete arc. G(D1) ≅ S4, G(D2) ≅ Z2, G(D3) ≅ Z2, G(D4) ≅ I, G(D5) ≅ Z2, G(D6) ≅ I. The Classification of (8,3)-Arcs: There groups G(Di), i = 1, …, 6 partition the points of index zero for Di into 239 orbits and so there exist 239 (8,3)-arcs to be constructed and by projectively equivalent of arcs, we have only seven projectively distinct (8,3)-arcs which are: E1 = {1,2,11,21,3,12,20,42}, E2 = {1,2,11,21,3,12,20,32}, E3 = {1,2,11,21,3,12,23,34}, E4 = {1,2,11,21,3,12,33,45}, E5 = {1,2,11,21,3,13,32,53}, E6 = {1,2,11,21,3,32,40,54}, E7 = {1,2,11,21,4,32,42,53}. G(E1) ≅ Z2, G(E2), G(E3), G(E4), G(E5), G(E6) and G(E7) are isomorphic to the identity group. Each one of these arcs is incomplete. The Classification of (9,3)-Arcs: There groups G(Ei) partition the points of index zero for Ei i = 1, …, 7 into 369 orbits. So there exist 369 (9,3)-arcs to be constructed and by projectively equivalent of arcs, we have only eight projectively distinct (9,3)-arcs which are: F1 = {1,2,11,21,3,12,20,42,43}, F2 = {1,2,11,21,3,12,20,42,33}, F3 = {1,2,11,21,3,12,20,42,32}, F4 = {1,2,11,21,3,12,20,32,40}, F5 = {1,2,11,21,3,12,23,34,45}, F6 = {1,2,11,21,3,12,33,45,79}, F7 = {1,2,11,21,3,32,40,54,37}, F8 = {1,2,11,21,3,32,40,54,79}. Each one is an incomplete arc. 268 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 The groups G(F1), G(F2), G(F3), G(F5), G(F7) and G(F8) consist of the identity. G(F4) ≅ Z2, G(F6) ≅ Z3. The Classification of (10,3)-Arcs: There groups G(Fi), i = 1, …, 8 partition the points of index zero for Fi into 307 orbits. So there exist 307 (10,3)-arcs and by projectively equivalent of arcs, we have only nine (10,3)-arcs which are projectively distinct and each one of them is an incomplete arc H1 = {1,2,11,21,3,12,20,42,43,69}, H2 = {1,2,11,21,3,12,20,42,43,33}, H3 = {1,2,11,21,3,12,20,42,43,35}, H4 = {1,2,11,21,3,12,20,42,43,32}, H5 = {1,2,11,21,3,12,20,42,33,86}, H6 = {1,2,11,21,3,12,20,32,40,54}, H7 = {1,2,11,21,3,12,23,34,45,60}, H8 = {1,2,11,21,3,32,40,54,37,15}, H9 = {1,2,11,21,3,32,40,54,37,79}. G(H1) ≅ Z3, G(H2), G(H3), G(H4), G(H5), G(H6), G(H7) and G(H8) are isomorphic to the identity group G(H9) ≅ Z2. The Classification of (11,3)-Arcs: There groups G(Hi), i = 1, …, 9 partition the points of index zero for Hi into 258 orbits. So we have 258 (11,3)-arcs to be constructed. By projectively equivalent of arcs, we have only nine (11,3)-arcs which are projectively distinct : K1 = {1,2,11,21,3,12,20,42,43,69,34}, K2 = {1,2,11,21,3,12,20,42,43,69,32}, K3 = {1,2,11,21,3,12,20,42,43,33,32}, K4 = {1,2,11,21,3,12,20,42,43,35,32}, K5 = {1,2,11,21,3,12,20,42,33,86,32}, K6 = {1,2,11,21,3,12,20,32,40,54,37}, K7 = {1,2,11,21,3,12,23,34,45,60,36}, K8 = {1,2,11,21,3,32,40,54,37,15,72}, K9 = {1,2,11,21,3,32,40,54,37,79,50}. The groups G(K1), G(K2), G(K3), G(k4), G(K5), G(K6), G(K8) and G(K9) are isomorphic to the identity group, the group G(K7) ≅ Z2. The Classification of (12,3)-Arcs: There groups G(Ki), i = 1, …, 9 partition the points of index zero for Ki into 196 orbits. So there exist 196 (12,3)-arcs to be constructed. By projectively equivalent of arcs, we have only nine projectively distinct (12,3)-arcs which are: L1 = {1,2,11,21,3,12,20,42,43,69,34,35}, L2 = {1,2,11,21,3,12,20,42,43,69,34,32}, L3 = {1,2,11,21,3,12,20,42,43,69,32,54}, L4 = {1,2,11,21,3,12,20,42,43,33,32,49}, L5 = {1,2,11,21,3,12,20,42,43,35,32,63}, L6 = {1,2,11,21,3,12,20,32,40,54,37,42}, L7 = {1,2,11,21,3,12,23,34,45,60,36,46}, L8 = {1,2,11,21,3,32,40,54,37,15,72,50}, L9 = {1,2,11,21,3,32,40,54,37,15,72,79}. Each one is an incomplete arc. The groups G(Li), i = 1, …, 9 are isomorphic to the identity group. The Classification of (13,3)-Arcs: There groups G(Li) partition the points of index zero for Li, i = 1, …, 9 into 137 orbits. So there exist 137 (13,3)-arcs to be constructed and by projectively equivalent of arcs, we have only nine projectively distinct (13,3)-arcs which are: M1 = {1,2,11,21,3,12,20,42,43,69,34,35,62}, M2 = {1,2,11,21,3,12,20,42,43,69,34,35,58}, M3 = {1,2,11,21,3,12,20,42,43,69,34,32,54}, M4 = {1,2,11,21,3,12,20,42,43,69,32,54,85}, 269 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 M5 = {1,2,11,21,3,12,20,42,43,33,32,49,86}, M6 = {1,2,11,21,3,12,20,42,43,35,32,63,64}, M7 = {1,2,11,21,3,12,20,32,40,54,37,42,53}, M8 = {1,2,11,21,3,12,23,34,45,60,36,46,49}, M9 = {1,2,11,21,3,32,40,54,37,15,72,50,79}. Each one is an incomplete arc. The groups G(Mi), i = 1, 2, 3, 5, …, 9 are isomorphic to the identity group G(M4) ≅ Z3. The Classification of (14,3)-Arcs: There groups G(Mi) partition the points of index zero for Mi, i = 1, …, 9 into eight projectively distinct (14,3)-arcs which are: N1={1,2,11,21,3,12,20,42,43,69,34,35,62,58}, N2= {1,2,11,21,3,12,20,42,43,69,34,35,58,68}, N3= {1,2,11,21,3,12,20,42,43,69,34,32,54,68}, N4={1,2,11,21,3,12,20,42,43,33,32,49,86,53}, N5={1,2,11,21,3,12,20,42,43,33,32,49,86,64}, N6={1,2,11,21,3,12,20,42,43,35,32,63,64,86}, N7={1,2,11,21,3,12,20,32,40,54,37,42,53,63}, N8= {1,2,11,21,3,32,40,54,37,15,72,50,79,22}. N1, N2, N3 are complete, while N4, …, N8 are incomplete arcs. The groups G(Ni), i = 1,2,…, 8 are isomorphic to the identity group. The Classification of (15,3)-Arcs: There groups G(Ni), i = 4, …, 8 partition the points of index zero for Ni, into 21 orbits. So there exist 21 (15,3)-arcs to be constructed and by projectively equivalent of arcs, we have only six projectively distinct (15,3)-arcs which are: Q1 = {1,2,11,21,3,12,20,42,43,33,32,49,86,53,70}, Q2 = {1,2,11,21,3,12,20,42,43,33,32,49,86,53,76}, Q3 = {1,2,11,21,3,12,20,42,43,33,32,49,86,64,70}, Q4 = {1,2,11,21,3,12,20,42,43,35,32,63,64,86,78}, Q5 = {1,2,11,21,3,12,20,32,40,54,37,42,53,63,68}, Q6 = {1,2,11,21,3,32,40,54,37,15,72,50,79,22,56}. The groups G(Qi), i = 1, …, 6 are isomorphic to the identity group. Q1, Q2, Q4 and Q5 are complete since there are no points of index zero for them. The groups G(Q3) and G(Q6) partition the points of index zero for Q3 and Q6 into 5 orbits. So, there exist five (16,3)-arcs and by projectively equivalent of arcs, we have only two projectively distinct (16,3)-arcs which are: R1 = {1,2,11,21,3,12,20,42,43,33,32,49,86,64,70,67}, R2 = {1,2,11,21,3,32,40,54,37,15,72,50,79,22,56,64}. R1 and R2 are complete (16,3)-arcs since C0 = 0 for R1 and R2. References 1.Hirschfeld, J. W. P. (1979) Projective Geometries Over Finite Fields, Second Edition, Oxford University Press. 2.Kareem, F.F. (2000) Classification and Construction of (k,3)-arcs in PG(2,9), M.Sc. Thesis, University of Baghdad. 3.Kwaam, A.A. (1999) Classification and Construction of (k,3)-arcs in PG(2,11), M.Sc. Thesis, University of Baghdad, Iraq. 270 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 (GF(9),∗) ∗ 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 1 6 8 7 3 5 4 3 3 6 4 7 1 8 2 5 4 4 8 7 2 3 5 6 1 5 5 7 1 3 8 2 4 6 6 6 3 8 5 2 4 1 7 7 7 5 2 6 4 1 8 3 8 8 4 5 1 6 7 3 2 (GF(9),+) + 0 1 2 3 4 5 6 7 8 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 6 7 8 0 1 2 4 4 5 3 7 8 6 1 2 0 5 5 3 4 8 6 7 2 0 1 6 6 7 8 0 1 2 3 4 5 7 7 8 6 1 2 0 4 5 3 8 8 6 7 2 0 1 5 3 4 Points and Lines of PG(2,9) i Pi ℓi 1 (1,0,0) 2 11 20 29 38 47 56 65 74 83 2 (0,1,0) 1 11 12 13 14 15 16 17 18 19 3 (1,1,0) 4 11 22 30 44 55 63 68 9 87 4 (2,1,0) 3 11 21 31 41 51 61 71 81 91 5 (3,1,0) 9 11 27 34 40 53 60 66 82 86 6 (4,1,0) 6 11 24 37 45 49 59 70 80 84 7 (5,1,0) 8 11 26 32 46 52 58 69 75 90 8 (6,1,0) 7 11 25 36 39 50 64 67 78 89 9 (7,1,0) 5 11 23 35 42 54 57 73 76 88 10 (8,1,0) 10 11 28 33 43 48 62 72 77 85 11 (0,0,1) 1 2 3 4 5 6 7 8 9 10 12 (1,0,1) 2 13 22 31 40 49 58 67 76 85 13 (2,0,1) 2 12 21 30 39 48 57 66 75 84 14 (3,0,1) 2 18 27 36 45 54 63 72 81 90 15 (4,0,1) 2 15 24 33 42 51 60 69 78 87 16 (5,0,1) 2 17 26 35 44 53 62 71 80 89 17 (6,0,1) 2 16 25 34 43 52 61 70 79 88 18 (7,0,1) 2 14 23 32 41 50 59 68 77 86 19 (8,0,1) 2 19 28 37 46 55 64 73 82 91 20 (0,1,1) 1 29 30 31 32 33 34 35 36 37 21 (1,1,1) 4 13 21 29 46 54 62 70 78 86 22 (2,1,1) 3 12 22 29 42 52 59 72 82 89 23 (3,1,1) 9 18 25 29 44 51 58 73 77 84 271 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 24 (4,1,1) 6 15 28 29 40 50 63 71 75 88 25 (5,1,1) 8 17 23 29 43 49 64 66 81 87 26 (6,1,1) 7 16 27 29 41 55 57 69 80 85 27 (7,1,1) 5 14 26 29 45 48 60 67 79 91 28 (8,1,1) 10 19 24 29 39 53 61 68 76 90 29 (0,2,1) 1 20 21 22 23 24 25 26 27 28 30 (1,2,1) 3 13 20 30 43 50 60 73 80 90 31 (2,2,1) 4 12 20 31 45 53 64 69 77 88 32 (3,2,1) 7 18 20 34 46 48 59 71 76 87 33 (4,2,1) 10 15 20 37 44 52 57 67 81 86 34 (5,2,1) 5 17 20 32 39 51 63 70 82 85 35 (6,2,1) 9 16 20 36 42 49 62 68 75 91 36 (7,2,1) 8 14 20 35 40 55 61 72 78 84 37 (8,2,1) 6 19 20 33 41 54 58 66 79 89 38 (0,3,1) 1 74 75 76 77 78 79 80 81 82 39 (1,3,1) 8 13 28 34 45 51 57 68 74 89 40 (2,3,1) 5 12 24 36 43 55 58 71 74 86 41 (3,3,1) 4 18 26 37 42 50 61 66 74 85 42 (4,3,1) 9 15 22 35 41 48 64 70 74 90 43 (5,3,1) 10 17 25 30 40 54 59 69 74 91 44 (6,3,1) 3 16 23 33 46 53 63 67 74 84 45 (7,3,1) 6 14 27 31 39 52 62 73 74 87 46 (8,3,1) 7 19 21 32 44 49 60 72 74 88 47 (0,4,1) 1 47 48 49 50 51 52 53 54 55 48 (1,4,1) 10 13 27 32 42 47 64 71 79 84 49 (2,4,1) 6 12 25 35 46 47 60 68 81 85 50 (3,4,1) 8 18 24 30 41 47 62 67 82 88 51 (4,4,1) 4 15 23 34 39 47 58 72 80 91 52 (5,4,1) 7 17 22 33 45 47 61 73 75 86 53 (6,4,1) 5 16 28 31 44 47 59 66 78 90 54 (7,4,1) 9 14 21 37 43 47 63 69 76 89 55 (8,4,1) 3 19 26 36 40 47 57 70 77 87 56 (0,5,1) 1 65 66 67 68 69 70 71 72 73 57 (1,5,1) 9 13 26 33 39 55 59 65 81 88 58 (2,5,1) 7 12 23 37 40 51 62 65 79 90 59 (3,5,1) 6 18 22 32 43 53 57 65 78 91 60 (4,5,1) 5 15 27 30 46 49 61 65 77 89 61 (5,5,1) 4 17 28 36 41 52 60 65 76 84 62 (6,5,1) 10 16 21 35 45 50 58 65 82 87 63 (7,5,1) 3 14 24 34 44 54 64 65 75 85 64 (8,5,1) 8 19 25 31 42 48 63 65 80 86 65 (0,6,1) 1 56 57 58 59 60 61 62 63 64 66 (1,6,1) 5 13 25 37 41 53 56 72 75 87 67 (2,6,1) 8 12 27 33 44 50 56 70 76 91 68 (3,6,1) 3 18 28 35 39 49 56 69 79 86 69 (4,6,1) 7 15 26 31 43 54 56 68 82 84 70 (5,6,1) 6 17 21 34 42 55 56 67 77 90 71 (6,6,1) 4 16 24 32 40 48 56 73 81 89 72 (7,6,1) 10 14 22 36 46 51 56 66 80 88 272 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 73 (8,6,1) 9 19 23 30 45 52 56 71 78 85 74 (0,7,1) 1 38 39 40 41 42 43 44 45 46 75 (1,7,1) 7 13 24 35 38 52 63 66 77 91 76 (2,7,1) 9 12 28 32 38 54 61 67 80 87 77 (3,7,1) 10 18 23 31 38 55 60 70 75 89 78 (4,7,1) 8 15 21 36 38 53 59 73 79 85 79 (5,7,1) 3 17 27 37 38 48 58 68 78 88 80 (6,7,1) 6 16 26 30 38 51 64 72 76 86 81 (7,7,1) 4 14 25 33 38 49 57 71 82 90 82 (8,7,1) 5 19 22 34 38 50 62 69 81 84 83 (0,8,1) 1 83 84 85 86 87 88 89 90 91 84 (1,8,1) 6 13 23 36 44 48 61 69 82 83 85 (2,8,1) 10 12 26 34 41 49 63 73 78 83 86 (3,8,1) 5 18 21 33 40 52 64 68 80 83 87 (4,8,1) 3 15 25 32 45 55 62 66 76 83 88 (5,8,1) 9 17 24 31 46 50 57 72 79 83 89 (6,8,1) 8 16 22 37 39 54 60 71 77 83 90 (7,8,1) 7 14 28 30 42 53 58 70 81 83 91 (8,8,1) 4 19 27 35 43 51 59 67 75 83 273 | Mathematics @@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 GF(9)في مستوي إسقاطي حول حقل كالوا (k,3)تصنیف وبناء االقواس عادل محمود أحمد آمال شھاب المختار فاطمة فیصل كریم جامعة بغداد ) /ابن الھیثم(كلیة التربیة للعلوم الصرفة /الریاضیات علوم قسم 2001نیسان 23قبل البحث في : ، 2001 یارا 27استلم البحث في : الخالصة حیث أن PG(2,9)االسقاطیة المختلفة في المستوي االسقاطي (k,3)یتم في ھذا البحث، بناء وتصنیف االقواس k ≥ 5 وبرھان ان االقواس ،– (k,3) 5الكاملة غیر موجودة، حیث أن ≤ k ≤ 3 وقد وجد أن اكبر قوس– (k,3) كامل . (14,3)–كامل القوس (k,3) –وس وان اصغر ق (16,3) –ھو القوس PG(2,9)في الكاملة بینھما. (k,3) –عالوة على ذلك، تم ایجاد االقواس ،مستوي اسقاطي ،حقل كالوا أقواس الكلمات المفتاحیة : 274 | Mathematics Theorem 1: [2] Notation 2: [2] Definition 4: [2]: قسم علوم الرياضيات / كلية التربية للعلوم الصرفة (ابن الهيثم) / جامعة بغداد يتم في هذا البحث، بناء وتصنيف الاقواس (k,3) الاسقاطية المختلفة في المستوي الاسقاطي PG(2,9) حيث أن k ( 5، وبرهان ان الاقواس – (k,3) الكاملة غير موجودة، حيث أن 5 ( k ( 3 وقد وجد أن اكبر قوس – (k,3) كامل في PG(2,9) هو القوس – (16,3) وان ا... علاوة على ذلك، تم ايجاد الاقواس – (k,3) الكاملة بينهما.