IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 A Complete (k,r)-Cap in PG(3,p) Over Galo is Field GF(4) F. F. Kareem Departme nt of Mathematics, Ibn-Al-Haitham College of Education , Unive rsity of Baghdad Received in: 27, June , 2010 Accepte d in: 12, December, 2010 Abstract The aim of this p aper is t o construct t he (k,r)-caps in the projective 3-sp ace PG(3,p ) over Galois field GF(4). We found that the maximum comp lete (k,2)-cap which is called an ovaloid , exists in PG(3,4) when k = 13. M oreover the maximum (k,3)-cap s, (k,4)-cap s and (k,5)-cap s. Key words: Projective Sp ace M aximum Comp lete (k,r) –cap Galois Field Introduction M any of the researchers worked on the construction and classification of the (k,n)-arcs in the projective planes PG(2,P),2≤ P ≥17. Now,I st udy of a finite p roective sp aces PG(3,P) over Galois field GF(P),It is the largest of the p rojective p lane over Galois field,Hirschfeld, [1] give the basic definition and theorems of p rojective geometrics over finite fields, and Al- M ukhtar, A.SH. in [2] give the comp lete Arcs and surfaces in three dimensional p rojective sp ace over Galois field and give (k,r)-cap s in PG(3,q) over Galois fields GF(q), q = 2,3, and 5. In this work we construct the (k,r)-cap s in PG(3,4). This p aper is divided into six sections, section one is the p reliminaries of p rojective 3-sp ace which contains some definitions and theorems for that concept and section two consists of the additions and multiplications op erations of GF(4). In section three to section six the construction of maximum comp lete (k,r)-cap s for r = 2,3,4,5.This work I have done manually without using the comp uter p rogram. 1- Preliminaries 1.1 De fini tion: "Projective 3-S pace", [3] A p rojective 3-sp ace PG(3,k) over a field k is a 3-dimensional p rojective sp ace which consists of p oints, lines and p lanes with the incidence relation between them. The p rojective 3-sp ace satisfies the following axioms: A) Any two dist inct p oints are contained in a unique line. B) Any three distinct non-collinear p oints, also any line and p oint not on the line are contained in a unique p lane. C) Any two dist inct coplanar lines intersect in a unique p oint. D) Any line not on a given p lane intersects t he plane in a unique point. E) Any two dist inct p lanes intersect in a unique line. A p rojective sp ace PG(3,p ) over Galois field GF(p ) , p = q m for some p rime number q and some integer m, is a 3-dimensional p rojective sp ace. Any p oint in PG(3,p ) has the form of a quadrable (x1,x2,x3,x4), where x1,x2,x3,x4 are elements in GF(p ) with the excep tion of the quadrable consisting of four zero elements. Two aquadrables (x1,x2,x3,x4) and (y 1,y 2,y 3,y 4) represent the same p oint if there exists  in GF(p )\{0} such that (x1,x2,x3,x4) =  (y 1,y 2,y 3,y 4). Similarly , any p lane in PG(3,p ) has the form of a quadrable [x1,x2,x3,x4], where x1,x2,x3,x4 are elements in GF(p ) with the excep tion of the quadrable consisting of four zero elements. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 Two quadrables [x1,x2,x3,x4] and [y 1,y 2,y 3,y 4] represent the same p lane if there exists  in GF(p )\{0} such that [x1,x2,x3,x4] =  [y 1,y 2,y 3,y 4]. Also a p oint p (x1,x2,x3,x4) is in cident with the p lane  [a1,a2,a3,a4] iff a1x1 + a2x2 + a3x3 + a4x4 = 0. 1.2 De fini tion: "Plan ", [1] A p lan  in PG(3,p ) is the set of all p oints p (x1,x2,x3,x4) satisfy ing a linear equation u1x1 + u2x2 + u3x3 + u4x4 = 0. This p lane is denoted by  [u1,u2,u3,u4]. 1.3 The orem: [2] Four dist inct p oints A(x1,x2,x3,x4), B(y 1,y 2,y 3,y 4), C(z1,z2,z3,z4) and D(w1,w2,w3,w4) are coplanar iff 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 x x x x y y y y 0 z z z z w w w w    . 1.4 Corollary: [2] If four distinct p oints A(x1,x2,x3,x4), B(y 1,y 2,y 3,y 4), C(z1,z2,z3,z4) and D(w1,w2,w3,w4) are collinear, then  = 0. 1.5 The orem: [2] The p oints of PG(3,p ) have unique forms which are (1,0,0,0), (x,1,0,0), (x,y ,1,0), (x,y ,z,1) for all x, y , z in GF(p ). 1.6 The orem: [2] The p lanes of PG(3,p ) have unique forms which are [1,0,0,0], [x,1,0,0], [x,y ,1,0], [x,y ,z,1] for all x, y , z in GF(p ). 1.7 The orem: [2] A p rojective 3-sp ace PG(3,p ) satisfies t he following: A) Every line contains exactly p + 1 p oints and every p oint is on exactly p + 1 lines. B) Every p lane contains exactly p 2 + p + 1 p oints (lines) and every p oint is on exactly p 2 + p + 1 p lanes. C) There exists p 3 + p 2 + p + 1 of p oints and there exists p 3 + p 2 + p + 1 of p lanes. D) Any two p lanes intersect in exactly p + 1 p oints and any line is on exactly p + 1 p lanes, and any two p oints are on exactly p + 1 p lanes. 1.8 The orem: [2] There exists (p 2 +1)(p 2 +p +1) of lines in PG(3,P). 1.9 De fini tion: [2] A (k,ℓ)-set in PG(3,p ) is a set of k sp aces ℓ. A k-set is a (k,0)-set that is a set of k-p oints. 1.10 De fini tion: "(k,r)-cap", [1] A (k,r)-cap is a set of k p oints in PG(n,p ) with n  3, such that at most r p oints on any line. Thus (k,2)-cap is a set of k p oints in PG(3,p ), such that no t hree of them are collinear. 1.11 De fini tion: "Comple te (k,r)-cap", [2] A (k,r)-cap is a complete if it is not contained in a (k+1,r)-cap. 1.12 De fini tion: [2] Let Ci be the number of p oints of index i in PG(3,p ) which are not on a (k,r)-cap then the constants Ci of (k,r)-cap satisfy the following: i) 3 2 iC p p p 1 k        ii) 2 i k(k 1)...(k n 1) iC (p p 1 n) n!          where  is the smallest i for which Ci 0,  be the largest i for which Ci 0. 1.13 Remark: [3] The (k,r)-cap is comp lete iff C0 = 0. 1.14 De fini tion: [2] IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 The i-secant of a (k,r)-cap is a line intersects the cap in exactly i points, that is 0-secant is an external line, 1-secant is a unisecant line, 2-secant is a bisecant line and 3-secant is atrisecant line. 1.15 Remark:[3] A (k,r)-cap is maximum iff every line in PG(3,p ) is a 0-secant or r-secant. 1.16 The orem: [1] A maximum (k,2)-cap in PG(3,p ) is an ovaloid. 2- The Additions and M ultip lications Op eration of GF(4): [4] To find the addition and multiplication tables in GF(4), we have the order p airs (x1,x2) such that x1, x2 in GF(2), as follows: 0  (0,0), 1  (1,0), 2  (0,1), 3  (1,1) Put these points in one orbit, (1,0) at the first p oint and by the principle of (1,0) A i , i = 0,1,2,3 and 0 1 1 1         , (1,0)A  (0,1) and (1,0)A 2  (1,1), so 0 1 (0,1) (1, 0) 1 1 (1,1)        . Now, in the left of the following table, m is the op eration of multiplication and in the right n is the operation of addition in multiplication side we write the numeration of p oints as last , and the addition side takes t he normal sequence. m(*) (+)n = f(m) 1 (1,0) 0 2 (0,1) 1 3 (1,1) 2 mod 3 In addition table, we have the following relation: (x1,x2) + (y 1,y 2) = (z 1,z2) where zi = (y i + xi) mod (2), for i = 1, 2. In multiplication table, we have the following relation ( (1,0) A f(m 1 ) ) A f(m 2 )  m1*m2 = m3 = (1,0) A (f (m 1 )+ f( m 2 )) (m od 3) = (x1,x2) For examp le: 23=1((1,0)A 1 )A 2 = (1,0)A 3 = (1,0)A 0 = (1,0) where (1,0) equal to 1 in multip lication side. Now we have addition and multiplication tables: + 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 3- The (k,2)-caps i n PG(3,4): PG(3,4) contains 85 p oints and 85 planes such that each p oint is on 21 p lanes and every p lane contains 21 p oints and every line contains 5 p oints and it is the intersection of 5 p lanes, (table 1 and 2). In table (1), the set A={1,2,6,22,43} is t aken which is t he set of unit and reference p oints 1(1,0,0,0), 2(0,1,0,0), 6(0,0,1,0), 22(0,0,0,1), 43(1,1,1,1), this set is a (5,2)-cap since no three p oints of A are collinear as in table (2). * 1 2 3 1 1 2 3 2 2 3 1 3 3 1 2 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 A is a (5,2)-cap , which is not comp lete since there exists some p oint of index zero for it, which are (12,13,15,16,17,19,20,21,28,29,31,32,33,35,36,37,40,41,46,48,49,50,52,53,55,56, 57,58,60, 1,62,63,65,66,67,68,69,71,72,73,74,76,77,78,79,80,81,82,83,84), then one can add some of them to A in order to obtain a comp lete (13,2)-cap B; B=A{12,15,21,28,31,37,40,46}={1,2,6,12,15,21,22,28,31,37,40,43,46}, B is the maximum (13,2)-cap in PG(3,4), since every line is a 0-secant or 2-secant, B is called an ovaloid. 4- The (k,3)-caps i n PG(3,4): Let B={1,2,6,12,15,21,22,28,31,37,40,43,46} be a (13,2)-cap . The p oints of index zero are (2,4,5,7,8,9,10,11,12,13,14,16,17,18,19,20,23,24,25,26,27,29,30,32,33,34,35,36,38,39,41, 42,44,45,47,…,85). The distinct (k,3)-cap can be constructed by adding some points of index zero for B, which are 3,7,10,23,26,39,42,54,57,61,64,66,67. Then C=B{3,7,10,23,26,39,42, 54,57,61,64,66,67}={1,2,3,6,7,10,12,15,21,22,23,26,28,31,37,39,40,42,43,46,54,57,61,64,66, 67},C is comp lete (26,3)-cap , since there are no p oints of index zero, i.e. C0=0. B is a maximum comp lete (k,3)-cap. 5- The (k,4)-caps i n PG(3,4): We can construct comp lete (k,4)-cap s by adding some p oints of index zero for C which are (4,5,8,9,11,13,14,16,17,18,19,20,24,25,27,29,30,32,33,34,35,36,38,41,44,45,47,48,49,50, 51,52,53,55,56,58,59,60,62,63,65,68,…,85), by adding to C nineteen of these p oints which are 4,9,11,14,17,24,27,33,38,44,47,48,55,59,60,62,65,68,74. Thus can get a comp lete (k,4)- cap call D={1,2,3,4,6,7,9,10,11,12,14,15,17,21,22,23,24,26,27,28,31,33,37,38,39,40,42,43, 44,46,47,48,54,55,57,59,60,62,64,65,66,67,68,74}. D is the maximum comp lete (45,4)-cap. 6- The (k,5)-caps i n PG(3,4): In section five, D is a comp lete (45,4)-cap. The points of index zero for D are (5,8,13,16, ,18,19,20,25,29,30,32,34,35,36,41,45,49,50,51,52,53,56,58,63,69,70,71,72,73,75,…,85), all of these p oints can be added to D, then E=D{5,8,13,16,18,19,20,25,29,30,32,34,36,41,45, 49,50,51,52,53,56,58,63,69,70,71,72,73,75,…,85} is the whole sp ace PG(3,4). E is the maximum comp lete (85,5)-cap which can be obtained for any line of PG(3,4) contains five p oints and hence there are no more than five are collinear. Conclusions; From the above results, t he dist inct comp lete (K,n)-cap s in PG(3,4),2≤ n≤ 5 Is as follows: (k,2)-cap , where k=13, is a complete maximum cap which is ovaloid. (k,3)-cap , where k=26, is a complete maximum cap . (k,4)-cap , where k=45, is a complete maximum cap . (k,5)-cap , where k=85, is a complete maximum cap , which is t he whole sp ace PG(3,4). Re ferences 1. Hirschfeld, J.W.P. (1998), Projective Geometries over Finite Fields, Second Edition, Oxford, University Press. 2. Al-M ukhtar, A. SH., (2008), Complete Arcs and Surfaces in Three Dimensional Projective Sp ace Over Galois Field, Thesis, University of Technology , Iraq. 3. Frank Ay res, JR., (1967), Projective Geometry , Schaum Publishing, Co. New York. 4. Hassan A. S., (2001), Const ruction of (k,3) – arcs on Projective Plane Over Galois Field GF(q), q = p h when p = 2 and h = 2, 3 and 4, M .Sc. Thesis, College of Education, Ibn-Al- Haitham,University of Baghdad. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 Table (1) Points and Plans of PG (3 , 4 ) i Pi Πi 1 (1,0,0,0) 2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 2 (0,1,0,0) 1 6 7 8 9 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 3 (1,1,0,0) 3 6 11 16 21 22 26 30 34 39 43 47 51 56 60 64 68 73 77 81 85 4 (2,1,0,0) 5 6 13 15 20 22 26 30 34 41 45 49 53 55 59 63 67 72 76 80 84 5 (3,1,0,0) 4 6 12 17 19 22 26 30 34 40 44 48 52 57 61 65 69 71 75 79 83 6 (0,0,1,0) 1 2 3 4 5 22 23 24 25 38 39 40 41 54 55 56 57 70 71 72 73 7 (1,0,1,0) 2 7 11 15 19 22 27 32 37 38 43 48 53 54 59 64 69 70 75 80 85 8 (2,0,1,0) 2 9 13 17 21 22 29 31 36 38 45 47 52 54 61 63 68 70 77 79 84 9 (3,0,1,0) 2 8 12 16 20 22 28 33 35 38 44 49 51 54 60 65 67 70 76 81 83 10 (0,1,1,0) 1 10 11 12 13 22 23 24 25 42 43 44 45 62 63 64 65 82 83 84 85 11 (1,1,1,0) 3 7 10 17 20 22 27 32 37 39 42 49 52 56 61 62 67 73 76 79 82 12 (2,1,1,0) 5 9 10 16 19 22 29 31 36 41 42 48 51 55 60 62 69 72 75 81 82 13 (3,1,1,0) 4 8 10 15 21 22 28 33 35 40 42 47 53 57 59 62 68 71 77 80 82 14 (0,2,1,0) 1 18 19 20 21 22 23 24 25 46 47 48 49 66 67 68 69 74 75 76 77 15 (1,2,1,0) 4 7 13 16 18 22 27 32 37 40 45 46 51 57 60 63 66 71 74 81 84 16 (2,2,1,0) 3 9 12 15 18 22 29 31 36 39 44 46 53 56 59 65 66 73 74 80 83 17 (3,2,1,0) 5 8 11 17 18 22 28 33 35 41 43 46 52 55 61 64 66 72 74 79 85 18 (0,3,1,0) 1 14 15 16 17 22 23 24 25 50 51 52 53 58 59 60 61 78 79 80 81 19 (1,3,1,0) 5 7 12 14 21 22 27 32 37 41 44 47 50 55 58 65 68 72 77 78 83 20 (2,3,1,0) 4 9 11 14 20 22 29 31 36 40 43 49 50 57 58 64 67 71 76 78 85 21 (3,3,1,0) 3 8 13 14 19 22 28 33 35 39 45 48 50 56 58 63 69 73 75 78 84 22 (0,0,0,1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 (1,0,0,1) 2 6 10 14 18 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 24 (2,0,0,1) 2 6 10 14 18 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 25 (3,0,0,1) 2 6 10 14 18 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 26 (0,0,1,1) 1 2 3 4 5 26 27 28 29 42 43 44 45 58 59 60 61 74 75 76 77 27 (1,0,1,1) 2 7 11 15 19 23 26 33 36 39 42 49 52 55 58 65 68 71 74 81 84 28 (2,0,1,1) 2 9 13 17 21 25 26 32 35 41 42 48 51 57 58 64 67 73 74 80 83 29 (3,0,1,1) 2 8 12 16 20 24 26 31 37 40 42 47 53 56 58 63 69 72 74 79 85 30 (0,0,2,1) 1 2 3 4 5 34 35 36 37 50 51 52 53 66 67 68 69 82 83 84 85 31 (1,0,2,1) 2 8 12 16 20 23 29 32 34 39 45 48 50 55 61 64 66 71 77 80 82 32 (2,0,2,1) 2 7 11 15 19 25 28 31 34 41 44 47 50 57 60 63 66 73 76 79 82 33 (3,0,2,1) 2 9 13 17 21 24 27 33 34 40 43 49 50 56 59 65 66 72 75 81 82 34 (0,0,3,1) 1 2 3 4 5 30 31 32 33 46 47 48 49 62 63 64 65 78 79 80 81 35 (1,0,3,1) 2 9 13 17 21 23 28 30 37 39 44 46 53 55 60 62 69 71 76 78 85 36 (2,0,3,1) 2 8 12 16 20 25 27 30 36 41 43 46 52 57 59 62 68 73 75 78 84 37 (3,0,3,1) 2 7 11 15 19 24 29 30 35 40 45 46 51 56 61 62 67 72 77 78 83 38 (0,1,0,1) 1 6 7 8 9 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 39 (1,1,0,1) 3 6 11 16 21 23 27 31 35 38 42 46 50 57 61 65 69 72 76 80 84 40 (2,1,0,1) 5 6 13 15 20 25 29 33 37 38 42 46 50 56 60 64 68 71 75 79 83 41 (3,1,0,1) 4 6 12 17 19 24 28 32 36 38 42 46 50 55 59 63 67 73 77 81 85 42 (0,1,1,1) 1 10 11 12 13 26 27 28 29 38 39 40 41 66 67 68 69 78 79 80 81 43 (1,1,1,1) 3 7 10 17 20 23 26 33 36 38 43 48 53 57 60 63 66 72 77 78 83 44 (2,1,1,1) 5 9 10 16 19 25 26 32 35 38 45 47 52 56 59 65 66 71 76 78 85 45 (3,1,1,1) 4 8 10 15 21 24 26 31 37 38 44 49 51 55 61 64 66 73 75 78 84 46 (0,1,2,1) 1 14 15 16 17 34 35 36 37 38 39 40 41 62 63 64 65 74 75 76 77 47 (1,1,2,1) 3 8 13 14 19 23 29 32 34 38 44 49 51 57 59 62 68 72 74 79 85 48 (2,1,2,1) 5 7 12 14 21 25 28 31 34 38 43 48 53 56 61 62 67 71 74 81 84 49 (3,1,2,1) 4 9 11 14 20 24 27 33 34 38 45 47 52 55 60 62 69 73 74 80 83 50 (0,1,3,1) 1 18 19 20 21 30 31 32 33 38 39 40 41 58 59 60 61 82 83 84 85 51 (1,1,3,1) 3 9 12 15 18 23 28 30 37 38 45 47 52 57 58 64 67 72 75 81 82 52 (2,1,3,1) 5 8 11 17 18 25 27 30 36 38 44 49 51 56 58 63 69 71 77 80 82 53 (3,1,3,1) 4 7 13 16 18 24 29 30 35 38 43 48 53 55 58 65 68 73 76 79 82 54 (0,2,0,1) 1 6 7 8 9 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 55 (1,2,0,1) 4 6 12 17 19 23 27 31 35 41 45 49 53 56 60 64 68 70 74 78 82 56 (2,2,0,1) 3 6 11 16 21 25 29 33 37 40 44 48 52 55 59 63 67 70 74 78 82 57 (3,2,0,1) 5 6 13 15 20 24 28 32 36 39 43 47 51 57 61 65 69 70 74 78 82 58 (0,2,1,1) 1 18 19 20 21 26 27 28 29 50 51 52 53 62 63 64 65 70 71 72 73 59 (1,2,1,1) 4 7 13 16 18 23 26 33 36 41 44 47 50 56 61 62 67 70 75 80 85 60 (2,2,1,1) 3 9 12 15 18 25 26 32 35 40 43 49 50 55 60 62 69 70 77 79 84 61 (3,2,1,1) 5 8 11 17 18 24 26 31 37 39 45 48 50 57 59 62 68 70 76 81 83 62 (0,2,2,1) 1 10 11 12 13 34 35 36 37 46 47 48 49 58 59 60 61 70 71 72 73 63 (1,2,2,1) 4 8 10 15 21 23 29 32 34 41 43 46 52 56 58 63 69 70 76 81 83 64 (2,2,2,1) 3 7 10 17 20 25 28 31 34 40 45 46 51 55 58 65 68 70 75 80 85 65 (3,2,2,1) 5 9 10 16 19 24 27 33 34 39 44 46 53 57 58 64 67 70 77 79 84 66 (0,2,3,1) 1 14 15 16 17 30 31 32 33 42 43 44 45 66 67 68 69 70 71 72 73 67 (1,2,3,1) 4 9 11 14 20 23 28 30 37 41 42 48 51 56 59 65 66 70 77 79 84 68 (2,2,3,1) 3 8 13 14 19 25 27 30 36 40 42 47 53 55 61 64 66 70 76 81 83 69 (3,2,3,1) 5 7 12 14 21 24 29 30 35 39 42 49 52 57 60 63 66 70 75 80 85 70 (0,3,0,1) 1 6 7 8 9 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 71 (1,3,0,1) 5 6 13 15 20 23 27 31 35 40 44 48 52 54 58 62 66 73 77 81 85 72 (2,3,0,1) 4 6 12 17 19 25 29 33 37 39 43 47 51 54 58 62 66 72 76 80 84 73 (3,3,0,1) 3 6 11 16 21 24 28 32 36 41 45 49 53 54 58 62 66 71 75 79 83 74 (0,3,1,1) 1 14 15 16 17 26 28 29 45 46 47 48 49 54 55 56 57 82 83 84 85 75 (1,3,1,1) 5 7 12 14 21 23 26 33 36 40 45 46 51 54 59 64 69 73 76 79 82 76 (2,3,1,1) 4 9 11 14 20 25 26 32 35 39 44 46 53 54 61 63 68 72 75 81 82 77 (3,3,1,1) 3 8 13 14 19 24 26 31 37 41 43 46 52 54 60 65 67 71 77 80 82 78 (0,3,2,1) 1 18 19 20 21 34 35 36 37 42 43 44 45 54 55 56 57 78 79 80 81 79 (1,3,2,1) 5 8 11 17 18 23 29 32 34 40 42 47 53 54 60 65 67 73 75 78 84 80 (2,3,2,1) 4 7 13 16 18 25 28 31 34 39 42 49 52 54 59 64 69 72 77 78 83 81 (3,3,2,1) 3 9 12 15 18 24 27 33 34 41 42 48 51 54 61 63 68 71 76 78 85 82 (0,3,3,1) 1 10 11 12 13 30 31 32 33 50 51 52 53 54 55 56 57 74 75 76 77 83 (1,3,3,1) 5 9 10 16 19 23 28 30 37 40 43 49 50 54 61 63 68 73 74 80 83 84 (2,3,3,1) 4 8 10 15 21 25 27 30 36 39 45 48 50 54 60 65 67 72 74 79 85 85 (3,3,3,1) 3 7 10 17 20 24 29 30 35 41 44 47 50 54 59 64 69 71 74 81 84 Table (2 ) Plans and line s of PG (3 , 4 ) 1 2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 6 22 22 22 22 2 2 2 2 6 14 10 18 6 18 14 10 6 10 18 14 10 26 42 50 46 38 42 46 50 42 30 34 26 58 30 34 26 74 30 34 26 14 30 62 58 66 54 58 62 66 46 66 58 62 62 38 38 38 78 50 42 46 18 34 82 78 74 70 74 78 82 50 70 70 70 66 82 74 78 82 54 54 54 2 1 6 7 8 9 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 6 22 22 22 22 1 7 6 6 9 6 1 8 9 7 1 9 1 7 8 8 7 26 27 28 29 23 26 28 29 25 23 26 23 23 25 30 24 35 24 25 24 8 30 32 33 31 24 33 32 33 32 31 27 32 28 28 31 27 36 29 27 26 9 34 37 35 36 25 36 36 37 35 35 29 34 37 34 33 34 37 30 30 31 3 3 6 11 16 21 22 26 30 34 39 43 47 51 56 60 64 68 73 77 81 85 6 22 22 22 22 3 11 21 16 6 3 16 21 6 11 3 3 16 11 21 6 11 26 43 51 47 39 39 39 39 43 26 26 26 60 34 30 34 30 30 34 73 16 30 64 60 68 56 68 60 64 47 60 56 64 64 47 47 51 43 51 43 77 21 34 85 81 77 73 81 85 77 51 77 85 73 68 73 81 85 68 56 56 81 4 5 6 13 15 20 22 26 30 34 41 45 49 53 55 59 63 67 72 76 80 84 6 22 22 22 22 5 13 13 20 6 15 13 20 6 20 15 5 6 5 5 15 13 26 45 53 49 41 41 53 45 45 30 34 26 59 30 34 34 76 26 30 26 15 30 63 59 67 55 67 55 55 49 67 59 63 63 41 41 53 80 45 49 49 20 34 84 80 76 72 80 76 80 53 72 72 72 67 84 76 84 84 59 63 55 5 4 6 12 17 19 22 26 30 34 40 44 48 52 57 61 65 69 71 75 79 83 6 22 22 22 22 4 4 4 4 6 17 17 12 6 12 17 12 19 6 19 19 12 26 44 52 48 40 44 48 52 44 30 26 30 61 34 34 26 26 71 34 30 17 30 65 61 69 57 61 65 69 48 69 57 57 65 48 40 40 52 79 44 40 19 34 83 79 75 71 75 79 83 52 71 83 75 69 71 75 79 65 83 57 61 6 1 2 3 4 5 22 23 24 25 38 39 40 41 54 55 56 57 70 71 72 73 2 22 22 22 22 1 2 2 2 1 5 3 4 5 1 5 3 1 3 4 4 3 38 39 40 41 23 39 40 41 39 24 25 23 23 54 25 23 71 24 25 24 4 54 56 57 55 24 55 56 57 40 57 55 56 40 56 38 38 72 41 39 38 5 70 73 71 72 25 71 72 73 41 70 70 70 73 57 71 72 73 54 54 55 7 2 7 11 15 19 22 27 32 37 38 43 48 53 54 59 64 69 70 75 80 85 7 22 22 22 22 2 2 2 2 7 15 11 19 15 7 15 11 7 11 19 19 11 27 43 53 48 38 43 48 53 43 32 37 27 27 54 37 27 75 32 37 32 15 32 64 59 69 54 59 64 69 48 69 59 64 48 64 38 38 80 53 43 38 19 37 85 80 75 70 75 80 85 53 70 70 70 85 69 75 48 85 54 54 59 8 2 9 13 17 21 22 29 31 36 38 45 47 52 54 61 63 68 70 77 79 84 9 22 22 22 22 2 2 2 2 21 9 13 21 13 9 17 13 17 9 21 17 13 29 45 52 47 38 45 47 52 31 38 36 29 31 54 36 29 31 70 36 29 17 31 63 61 68 54 61 63 68 61 47 61 63 52 63 38 38 45 79 45 47 21 36 84 79 77 70 77 79 84 84 52 70 70 77 68 77 79 68 84 54 54 9 2 8 12 16 20 22 28 33 35 38 44 49 51 54 60 65 67 70 76 81 83 8 22 22 22 22 2 2 2 2 8 16 16 12 8 12 16 12 20 8 20 20 12 28 44 51 49 38 44 49 51 44 33 28 33 60 35 35 28 28 70 35 33 16 33 65 60 67 54 60 65 67 49 67 54 54 65 49 38 38 51 81 44 38 20 35 83 81 76 70 76 81 83 51 70 83 76 67 70 76 81 65 83 54 60 1 0 1 10 11 12 13 22 23 24 25 42 43 44 45 62 63 64 65 82 83 84 85 10 22 22 22 22 1 10 10 10 1 12 11 12 13 1 13 13 1 11 11 12 11 42 43 44 45 23 43 44 45 43 25 25 23 23 62 25 24 83 24 23 24 12 62 64 65 63 24 63 64 65 44 62 63 64 44 64 42 43 84 45 42 42 13 82 85 83 84 25 83 84 85 45 84 82 82 85 65 83 82 85 62 65 63 1 1 3 7 10 17 20 22 27 32 37 39 42 49 52 56 61 62 67 73 76 79 82 7 22 22 22 22 3 3 10 10 10 7 17 20 20 20 7 17 7 17 3 3 10 27 42 52 49 39 42 52 49 27 39 27 27 37 32 56 32 76 37 32 37 17 32 62 61 67 56 61 56 61 67 49 56 62 42 39 61 42 79 39 49 52 20 37 82 79 76 73 76 76 73 79 52 82 73 79 82 67 73 82 62 62 67 1 2 5 9 10 16 19 22 29 31 36 41 42 48 51 55 60 62 69 72 75 81 82 9 22 22 22 22 5 10 10 10 16 16 16 9 9 5 5 5 19 9 19 19 10 29 42 51 48 41 41 51 48 36 31 29 41 60 29 31 36 29 72 36 31 16 31 62 60 69 55 69 55 60 62 69 55 42 62 42 48 51 51 81 42 41 19 36 82 81 75 72 81 75 72 75 72 82 48 69 75 81 82 62 82 55 60 1 4 8 10 15 21 22 28 33 35 40 42 47 53 57 59 62 68 71 77 80 82 3 8 22 22 22 22 4 10 10 10 15 15 15 8 21 21 21 8 8 4 4 4 10 28 42 53 47 40 40 53 47 35 33 28 40 35 33 28 57 77 28 33 35 15 33 62 59 68 57 68 57 59 62 68 57 42 42 40 53 59 80 42 47 53 21 35 82 80 77 71 80 77 71 77 71 82 47 80 82 71 62 82 59 62 68 1 4 1 18 19 20 21 22 23 24 25 46 47 48 49 66 67 68 69 74 75 76 77 18 22 22 22 22 1 18 18 18 1 20 21 21 20 1 19 21 1 20 19 19 19 46 48 49 47 23 47 48 49 47 24 25 24 23 66 23 23 75 25 25 24 20 66 69 67 68 24 67 68 69 48 69 67 66 48 68 49 46 76 46 47 46 21 74 75 76 77 25 75 76 77 49 74 74 75 77 69 74 76 77 68 66 67 1 5 4 7 13 16 18 22 27 32 37 40 45 46 51 57 60 63 66 71 74 81 84 7 22 22 22 22 4 18 18 18 7 16 16 13 7 13 16 13 7 4 4 4 13 27 45 51 46 40 51 40 45 45 32 27 32 60 37 37 27 74 27 32 37 16 32 63 60 66 57 63 60 57 46 66 57 57 63 46 40 40 81 45 46 51 18 37 84 81 74 71 71 84 81 51 71 84 74 66 71 74 81 84 60 63 66 1 6 3 9 12 15 18 22 29 31 36 39 44 46 53 56 59 65 66 73 74 80 83 9 22 22 22 22 3 18 18 18 15 15 15 9 12 12 9 12 9 3 3 3 12 29 44 53 46 39 53 39 44 36 31 29 39 31 36 56 29 74 29 31 36 15 31 65 59 66 56 65 59 56 65 66 56 44 53 46 59 39 80 44 46 53 18 36 83 80 74 73 73 83 80 74 73 83 46 74 73 66 80 83 59 65 66 1 7 5 8 11 17 18 22 28 33 35 41 43 46 52 55 61 64 66 72 74 79 85 8 22 22 22 22 5 11 11 11 18 18 8 18 8 5 17 5 17 8 5 17 11 28 43 52 46 41 41 52 46 33 35 41 28 61 28 35 35 33 72 33 28 17 33 64 61 66 55 66 55 61 61 55 43 64 64 43 41 52 43 79 46 46 18 35 85 79 74 72 79 74 72 85 79 52 72 66 74 74 85 66 85 64 55 1 8 1 14 15 16 17 22 23 24 25 50 51 52 53 58 59 60 61 78 79 80 81 14 22 22 22 22 1 14 14 14 1 17 15 16 1 16 17 15 1 15 16 17 15 50 53 51 52 23 51 52 53 51 25 23 24 59 25 23 24 79 25 23 24 16 58 59 60 61 24 59 60 61 52 58 58 58 60 52 53 51 80 50 50 50 17 78 80 81 79 25 79 80 81 53 80 81 79 61 78 78 78 81 60 61 59 1 9 5 7 12 14 21 22 27 32 37 41 44 47 50 55 58 65 68 72 77 78 83 7 22 22 22 22 5 14 14 14 21 21 5 7 12 7 21 5 12 5 12 7 12 27 44 50 47 41 47 44 41 32 37 32 41 32 55 27 37 37 27 27 72 14 32 65 58 68 55 55 68 65 58 55 65 44 50 65 50 50 47 44 41 77 21 37 83 78 77 72 83 72 77 83 78 78 47 77 68 72 83 58 58 68 78 2 0 4 9 11 14 20 22 29 31 36 40 43 49 50 57 58 64 67 71 76 78 85 9 22 22 22 22 4 14 14 14 20 20 11 9 9 4 4 4 20 11 11 9 11 29 43 50 49 40 49 43 40 31 36 36 40 58 29 31 36 29 31 29 71 14 31 64 58 67 57 57 67 64 58 57 58 43 64 43 49 50 50 50 40 76 20 36 85 78 76 71 85 71 76 85 78 71 49 67 76 78 85 64 57 67 78 2 1 3 8 13 14 19 22 28 33 35 39 45 48 50 56 58 63 69 73 75 78 84 8 22 22 22 22 3 3 3 13 14 14 8 13 14 19 8 13 19 18 19 3 13 28 45 50 48 39 45 48 48 28 33 39 33 28 33 56 28 28 73 35 35 14 33 63 58 69 56 58 63 58 35 69 45 56 48 39 58 39 50 78 45 50 19 35 84 78 75 73 75 78 73 63 73 50 75 84 84 69 78 63 84 56 69 2 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2 6 6 6 6 1 2 2 2 1 5 3 4 1 4 5 3 1 3 4 5 3 10 11 12 13 7 11 12 13 11 8 9 7 15 8 9 7 19 8 9 7 4 14 16 17 15 8 15 16 17 12 17 15 16 16 10 10 10 20 13 11 12 5 18 21 19 20 9 19 20 21 13 18 18 18 17 21 19 20 21 14 14 14 2 3 2 6 10 14 18 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 6 23 23 23 23 2 10 10 10 14 14 6 18 18 18 6 2 6 2 2 14 10 27 43 51 47 39 39 51 47 35 31 39 27 35 31 55 35 75 27 31 27 14 31 63 59 67 55 67 55 59 63 67 43 63 43 39 59 51 79 43 47 55 18 35 83 79 75 71 79 75 71 75 71 51 71 79 83 67 83 83 59 63 47 2 4 2 6 10 14 18 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 6 25 25 25 25 2 10 10 10 18 18 6 18 14 2 6 2 14 14 2 6 10 29 45 53 49 41 41 53 49 33 37 41 29 29 29 57 37 33 37 33 73 14 33 65 61 69 57 69 57 61 61 57 45 65 49 45 61 53 45 41 49 77 18 37 85 81 77 73 81 77 73 85 81 53 73 85 77 69 85 69 65 65 81 2 5 2 6 10 14 18 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 6 24 24 24 24 2 14 14 18 18 2 6 10 6 10 18 10 6 14 2 2 10 28 44 52 48 40 48 44 44 32 28 40 32 60 36 28 28 76 36 32 36 14 32 64 60 68 56 56 68 56 60 60 44 56 64 48 52 40 80 40 48 52 18 36 84 80 76 72 84 72 80 84 76 52 76 68 72 72 80 84 64 64 68 2 6 1 2 3 4 5 26 27 28 29 42 43 44 45 58 59 60 61 74 75 76 77 2 26 26 26 26 1 2 2 2 1 5 3 4 1 4 5 3 1 3 4 5 3 42 43 44 45 27 43 44 45 43 28 29 27 59 28 29 27 75 28 29 27 4 58 60 61 59 28 59 60 61 44 61 59 60 60 42 42 42 76 45 43 44 5 74 77 75 76 29 75 76 77 45 74 74 74 61 77 75 76 77 58 58 58 2 7 2 7 11 15 19 23 26 33 36 39 42 49 52 55 58 65 68 71 74 81 84 7 23 23 23 23 2 15 15 15 11 19 11 7 11 19 7 2 19 2 2 7 11 26 42 52 49 39 49 42 39 26 36 36 39 33 33 55 36 26 26 33 71 15 33 65 58 68 55 55 68 65 68 55 58 42 52 39 58 52 52 42 49 74 19 36 84 81 74 71 84 71 74 81 81 71 49 74 84 68 84 65 58 65 81 2 8 2 9 13 17 21 25 26 32 35 41 42 48 51 57 58 64 67 73 74 80 83 9 25 25 25 25 2 2 2 13 9 21 17 2 13 21 21 9 17 17 13 9 13 26 42 51 48 41 42 48 48 42 35 26 35 32 32 26 57 32 35 26 73 17 32 64 58 67 57 58 64 58 48 57 57 67 51 41 51 58 42 41 41 74 21 35 83 80 74 73 74 80 73 51 80 83 83 74 83 73 64 67 64 67 80 2 9 2 8 12 16 20 24 26 31 37 40 42 47 53 56 58 63 69 72 74 79 85 8 24 24 24 24 2 20 2 2 8 20 16 12 8 20 16 16 12 2 12 8 12 26 42 53 47 40 53 47 53 42 37 26 31 58 31 37 31 37 26 26 72 16 31 63 58 69 56 63 63 69 47 56 56 56 63 40 40 42 47 42 40 74 20 37 85 79 74 72 72 79 85 53 79 85 74 69 85 74 72 58 58 69 79 3 0 1 2 3 4 5 34 35 36 37 50 51 52 53 66 67 68 69 82 83 84 85 2 34 34 34 34 1 2 2 2 3 1 5 3 1 3 5 5 4 1 4 4 3 50 51 52 53 35 51 52 53 35 50 35 36 67 37 37 36 35 82 37 36 4 66 68 69 67 36 67 68 69 69 52 66 66 68 52 50 51 53 84 51 50 5 82 85 83 84 37 83 84 85 84 53 85 83 69 82 83 82 68 85 66 67 3 1 2 8 12 16 20 23 29 32 34 39 45 48 50 55 61 64 66 71 77 80 82 8 23 23 23 23 2 2 2 2 8 16 16 12 8 12 20 12 8 16 20 20 12 29 45 50 48 39 45 48 50 45 32 29 32 61 34 29 29 77 34 34 32 16 32 64 61 66 55 61 64 66 48 66 55 55 64 48 50 39 80 39 45 39 20 34 82 80 77 71 77 80 82 50 71 82 77 66 71 71 80 82 64 55 61 3 2 2 7 11 15 19 25 28 31 34 41 44 47 50 57 60 63 66 73 76 79 82 7 25 25 25 25 2 2 2 2 7 15 15 11 7 11 19 11 7 15 19 19 11 28 44 50 47 41 44 47 50 44 31 28 31 60 34 28 28 76 34 34 31 15 31 63 60 66 57 60 63 66 47 66 57 57 63 47 50 41 79 41 44 41 19 34 82 79 76 73 76 79 82 50 73 82 76 66 73 73 79 82 63 57 60 3 3 2 9 13 17 21 24 27 33 34 40 43 49 50 56 59 65 66 72 75 81 82 9 24 24 24 24 2 2 2 2 9 17 17 13 9 13 17 13 21 9 21 21 13 27 43 50 49 40 43 49 50 43 33 27 33 59 34 34 27 27 72 34 33 17 33 65 59 66 56 59 65 66 49 66 56 56 65 49 40 40 50 81 43 40 21 34 82 81 75 72 75 81 82 50 72 82 75 66 72 75 81 65 82 56 59 3 4 1 2 3 4 5 30 31 32 33 46 47 48 49 62 63 64 65 78 79 80 81 2 30 30 30 30 1 2 2 2 1 4 3 3 1 4 4 5 1 5 3 5 3 46 47 48 49 31 47 48 49 47 33 33 32 63 32 31 32 79 33 31 31 4 62 64 65 63 32 63 64 65 48 62 63 62 64 46 49 47 80 46 46 48 5 78 81 79 80 33 79 80 81 49 80 78 79 65 81 78 78 81 64 65 62 3 5 2 9 13 17 21 23 28 30 37 39 44 46 53 55 60 62 69 71 76 78 85 9 23 23 23 23 2 2 2 2 13 9 13 13 9 21 21 17 9 17 21 17 13 28 44 53 46 39 44 46 53 28 39 37 30 60 30 28 30 76 37 37 28 17 30 62 60 69 55 60 62 69 69 46 60 55 62 39 53 44 78 39 44 46 21 37 85 78 76 71 76 78 85 78 53 71 76 69 85 71 71 85 62 55 55 3 6 2 8 12 16 20 25 27 30 36 41 43 46 52 57 59 62 68 73 75 78 84 8 25 25 25 25 2 2 2 2 8 16 16 12 8 12 20 12 8 16 20 20 12 27 43 52 46 41 43 46 52 43 30 27 30 59 36 27 27 75 36 36 30 16 30 62 59 68 57 59 62 68 46 68 57 57 62 46 52 41 78 41 43 41 20 36 84 78 75 73 75 78 84 52 73 84 75 68 73 73 78 84 62 57 59 3 7 2 7 11 15 19 24 29 30 35 40 45 46 51 56 61 62 67 72 77 78 83 7 24 24 24 24 2 2 2 2 7 15 15 19 7 19 15 11 11 11 19 7 11 29 45 51 46 40 45 46 51 45 30 29 29 61 30 35 29 35 30 35 72 15 30 62 61 67 56 61 62 67 46 67 56 62 62 40 40 40 46 51 45 77 19 35 83 78 77 72 77 78 83 51 72 83 72 67 83 77 78 61 56 56 78 3 8 1 6 7 8 9 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 6 38 38 38 38 1 6 6 6 1 8 7 7 1 8 8 9 1 9 7 9 7 42 43 44 45 39 43 44 45 43 41 41 40 47 40 39 40 51 41 39 39 8 46 48 49 47 40 47 48 49 44 46 47 46 48 42 45 43 52 42 49 44 9 50 53 51 52 41 51 52 53 45 52 50 51 49 53 50 50 53 48 42 46 3 9 3 6 11 16 21 23 27 31 35 38 42 46 50 57 61 65 69 72 76 80 84 6 23 23 23 23 3 3 3 3 6 16 16 11 6 11 16 11 21 6 21 21 11 27 42 50 46 38 42 46 50 42 31 27 31 61 35 35 27 27 72 35 31 16 31 65 61 69 57 61 65 69 46 69 57 57 65 46 38 38 50 80 42 38 21 35 84 80 76 72 76 80 84 50 72 84 76 69 72 76 80 65 84 57 61 4 0 5 6 13 15 20 25 29 33 37 38 42 46 50 56 60 64 68 71 75 79 83 6 25 25 25 25 5 5 5 5 6 15 15 13 6 13 20 13 6 15 20 20 13 29 42 50 46 38 42 46 50 42 33 29 33 60 37 29 29 75 37 37 33 15 33 64 60 68 56 60 64 68 46 68 56 56 64 46 50 38 79 38 42 38 20 37 83 79 75 71 75 79 83 50 71 83 75 68 71 71 79 81 64 56 60 4 1 4 6 12 17 19 24 28 32 36 38 42 46 50 55 59 63 67 73 77 81 85 6 24 24 24 24 4 4 4 4 6 17 17 12 6 12 17 12 19 6 19 19 12 28 42 50 46 38 42 46 50 42 32 28 32 59 36 36 28 28 73 36 32 17 32 63 59 67 55 59 63 67 46 67 55 55 63 46 38 38 50 81 42 38 19 36 85 81 77 73 77 81 85 50 73 85 77 67 73 77 81 63 85 55 59 4 2 1 10 11 12 13 26 27 28 29 38 39 40 41 66 67 68 69 78 79 80 81 10 26 26 26 26 1 10 10 10 1 12 11 11 1 12 12 11 13 13 1 13 11 38 39 40 41 27 39 40 41 39 29 29 28 67 28 27 27 28 29 78 27 12 66 68 69 67 28 67 68 69 40 66 67 66 68 38 41 38 39 38 79 40 13 78 81 79 80 29 79 80 81 41 80 78 79 69 81 78 80 69 68 81 66 4 3 3 7 10 17 20 23 26 33 36 38 43 48 53 57 60 63 66 72 77 78 83 7 23 23 23 23 3 3 3 3 7 17 17 10 7 10 17 10 20 7 20 20 10 26 43 53 48 38 43 48 57 43 33 26 33 60 36 36 26 26 72 36 33 17 33 63 60 66 57 60 63 66 48 66 57 57 63 48 38 38 53 78 43 38 20 36 83 78 77 72 77 78 83 53 72 83 77 66 72 77 78 63 83 57 60 4 4 5 9 10 16 19 25 26 32 35 38 45 47 52 56 59 65 66 71 76 78 85 9 25 25 25 25 5 5 5 5 9 16 16 10 9 10 16 10 19 9 19 19 10 26 45 52 47 38 45 47 52 45 32 26 32 59 35 35 26 26 71 35 32 16 32 65 59 66 56 59 65 66 47 66 56 56 65 47 38 38 52 78 45 38 19 35 85 78 76 71 76 78 85 52 71 85 76 66 71 76 78 65 85 56 59 4 5 4 8 10 15 21 24 26 31 37 38 44 49 51 55 61 64 66 73 75 78 84 8 24 24 24 24 4 4 4 4 8 15 15 10 8 10 15 10 21 8 21 21 10 26 44 51 49 38 44 49 51 44 31 26 31 61 37 37 26 26 73 37 31 15 31 64 61 66 55 61 64 66 49 66 55 55 64 49 38 38 51 78 44 38 21 37 84 78 75 73 75 78 84 51 73 84 75 66 73 75 78 64 84 55 61 4 6 1 14 15 16 17 34 35 36 37 38 39 40 41 62 63 64 65 74 75 76 77 14 34 34 34 34 1 14 14 14 1 15 15 16 1 16 15 16 17 1 17 17 15 38 41 39 40 35 39 40 41 39 36 35 36 63 37 37 35 35 74 37 36 16 62 63 64 65 36 63 64 65 40 65 62 62 64 40 38 38 41 76 39 38 17 74 76 77 75 37 75 76 77 41 74 77 75 65 74 75 76 64 77 62 63 4 7 3 8 13 14 19 23 29 32 34 38 44 49 51 57 59 62 68 72 74 79 85 8 23 23 23 23 3 3 3 3 8 14 14 13 8 13 14 13 19 8 19 19 13 29 44 51 49 38 44 49 51 44 32 29 32 59 34 34 29 29 72 34 32 14 32 62 59 68 57 59 62 68 49 68 57 57 62 49 38 38 51 79 44 38 19 34 85 79 74 72 74 79 85 51 72 85 74 68 72 74 79 62 85 57 59 4 8 5 7 12 14 21 25 28 31 34 38 43 48 53 56 61 62 67 71 74 81 84 7 25 25 25 25 5 5 5 5 7 14 14 12 7 12 21 12 7 14 21 21 12 28 43 53 48 38 43 48 53 43 31 28 31 61 34 28 28 74 34 34 31 14 31 62 61 67 56 61 62 67 48 67 56 56 62 48 53 38 81 38 43 38 21 34 84 81 74 71 74 81 84 53 71 84 74 67 71 71 81 84 62 56 61 4 9 4 9 11 14 20 24 27 33 34 38 45 47 52 55 60 62 69 73 74 80 83 9 24 24 24 24 4 4 4 4 9 14 14 11 9 11 14 11 20 9 20 20 11 27 45 52 47 38 45 47 52 45 33 27 33 60 34 34 27 27 73 34 33 14 33 62 60 69 55 60 62 69 47 69 55 55 62 47 38 38 52 80 45 38 20 34 83 80 74 73 74 80 83 52 73 83 74 69 73 74 80 62 83 55 60 5 0 1 18 19 20 21 30 31 32 33 38 39 40 41 58 59 60 61 82 83 84 85 18 30 30 30 30 1 18 18 18 1 19 21 19 1 19 20 21 20 21 1 20 19 38 40 41 39 31 39 40 41 39 33 33 31 59 32 33 31 32 32 82 31 20 58 61 59 60 32 59 60 61 40 58 59 60 60 38 38 38 39 41 83 40 21 82 83 84 85 33 83 84 85 41 84 82 82 61 85 83 84 61 58 85 58 5 1 3 9 12 15 18 23 28 30 37 38 45 47 52 57 58 64 67 72 75 81 82 9 23 23 23 23 3 12 12 12 9 3 3 3 9 18 18 15 9 15 18 15 12 28 45 52 47 38 38 52 47 45 28 30 37 58 30 28 30 75 37 37 28 15 30 64 58 67 57 67 57 58 47 58 64 67 64 38 52 45 81 38 45 47 18 37 82 81 75 72 81 75 72 52 75 81 82 67 82 72 72 82 64 57 57 5 2 5 8 11 17 18 25 27 30 36 38 44 49 51 56 58 63 69 71 77 80 82 8 25 25 25 25 5 5 5 5 8 17 17 18 8 11 17 11 8 11 18 18 11 27 44 51 49 38 44 49 51 44 30 27 27 58 36 36 27 77 30 36 30 17 30 63 58 69 56 58 63 69 49 69 56 63 63 49 38 38 80 51 44 38 18 36 82 80 77 71 77 80 82 51 71 82 71 69 71 77 80 82 56 56 58 5 3 4 7 13 16 18 24 29 30 35 38 43 48 53 55 58 65 68 73 76 79 82 7 24 24 24 24 4 4 4 4 7 16 16 13 7 13 16 13 18 7 18 18 13 29 43 53 48 38 43 48 53 43 30 29 30 58 35 35 29 29 73 35 30 16 30 65 58 68 55 58 65 68 48 68 55 55 65 48 38 38 53 79 43 38 18 35 82 79 76 73 76 79 82 53 73 82 76 68 73 76 79 65 82 55 58 5 4 1 6 7 8 9 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 6 70 70 70 70 1 6 6 6 1 8 7 7 1 8 8 7 9 9 1 9 7 74 75 76 77 71 75 76 77 75 73 73 72 79 72 71 71 72 73 82 71 8 78 80 81 79 72 79 80 81 76 78 79 78 80 74 77 74 75 74 83 76 9 82 85 83 84 73 83 84 85 77 84 82 83 81 85 82 84 81 80 85 78 5 5 4 6 12 17 19 23 27 31 35 41 45 49 53 56 60 64 68 70 74 78 82 6 23 23 23 23 4 4 4 4 6 17 17 12 6 12 17 12 19 6 19 19 12 27 45 53 49 41 45 49 53 45 31 27 31 60 35 35 27 27 70 35 31 17 31 64 60 68 56 60 64 68 49 68 56 56 64 49 41 41 53 78 45 41 19 35 82 78 74 70 74 78 82 53 70 82 74 68 70 74 78 64 82 56 60 5 6 3 6 11 16 21 25 29 33 37 40 44 48 52 55 59 63 67 70 74 78 82 6 25 25 25 25 3 3 3 3 6 16 11 21 16 6 16 11 6 11 21 21 11 29 44 52 48 40 44 48 52 44 33 37 29 29 55 37 29 74 33 37 33 16 33 63 59 67 55 59 63 67 48 67 59 63 48 63 40 40 78 52 44 40 21 37 82 78 74 70 74 78 82 52 70 70 70 82 67 74 78 82 55 55 59 5 7 5 6 13 15 20 24 28 32 36 39 43 47 51 57 61 65 69 70 74 78 82 6 24 24 24 24 5 5 5 5 6 15 15 13 6 13 15 13 20 6 20 20 13 28 43 51 47 39 43 47 51 43 32 28 32 61 36 36 28 28 70 36 32 15 32 65 61 69 57 61 65 69 47 69 57 57 65 47 39 39 51 78 43 39 20 36 82 78 74 70 74 78 82 51 70 82 74 69 70 74 78 65 82 57 61 5 8 1 18 19 20 21 26 27 28 29 50 51 52 53 62 63 64 65 70 71 72 73 18 26 26 26 26 1 18 18 18 1 19 20 19 1 19 20 20 21 21 21 1 19 50 52 53 51 27 51 52 53 51 29 27 27 63 28 29 28 29 28 27 70 20 62 65 63 64 28 63 64 65 52 62 62 64 64 50 50 51 52 53 50 71 21 70 71 72 73 29 71 72 73 53 72 73 70 65 73 71 70 63 62 65 72 5 9 4 7 13 16 18 23 26 33 36 41 44 47 50 56 61 62 67 70 75 80 85 7 23 23 23 23 4 4 4 4 7 16 16 13 7 13 16 13 18 7 18 18 13 26 44 50 47 41 44 47 50 44 33 26 33 61 36 36 26 26 70 36 33 16 33 62 61 67 56 61 62 67 47 67 56 56 62 47 41 41 50 80 44 41 18 36 85 80 75 70 75 80 85 50 70 85 75 67 70 75 80 62 85 56 61 6 0 3 9 12 15 18 25 26 32 35 40 43 49 50 55 60 62 69 70 77 79 84 9 25 25 25 25 3 3 3 3 9 15 15 12 9 12 15 12 18 9 18 18 12 26 43 50 49 40 43 49 50 43 32 26 32 60 35 35 26 26 70 35 32 15 32 62 60 69 55 60 62 69 49 69 55 55 62 49 40 40 50 79 43 40 18 35 84 79 77 70 77 79 84 50 70 84 77 69 70 77 79 62 84 55 60 6 1 5 8 11 17 18 24 26 31 37 39 45 48 50 57 59 62 68 70 76 81 83 8 24 24 24 24 5 5 5 5 8 17 17 11 8 11 17 11 18 8 18 18 11 26 45 50 48 39 45 48 50 45 31 26 31 59 37 37 26 26 70 37 31 17 31 62 59 68 57 59 62 68 48 68 57 57 62 48 39 39 50 81 45 39 18 37 83 81 76 70 76 81 83 50 70 83 76 68 70 76 81 62 83 57 59 6 2 1 10 11 12 13 34 35 36 37 46 47 48 49 58 59 60 61 70 71 72 73 10 34 34 34 34 1 10 10 10 1 12 11 11 1 12 12 11 13 13 1 13 11 46 47 48 49 35 47 48 49 47 37 37 36 59 36 35 35 36 37 70 35 12 58 60 61 59 36 59 60 61 48 58 59 58 60 46 49 46 47 46 71 48 13 70 73 71 72 37 71 72 73 49 72 70 71 61 73 70 72 61 60 73 58 6 3 4 8 10 15 21 23 29 32 34 41 43 46 52 56 58 63 69 70 76 81 83 8 23 23 23 23 4 4 4 4 8 15 15 10 8 10 15 10 21 8 21 21 10 29 43 52 46 41 43 46 52 43 32 29 32 58 34 34 29 29 70 34 32 15 32 63 58 69 56 58 63 69 46 69 56 56 63 46 41 41 52 81 43 41 21 34 83 81 76 70 76 81 83 52 70 83 76 69 70 76 81 63 83 56 58 6 4 3 7 10 17 20 25 28 31 34 40 45 46 51 55 58 65 68 70 75 80 85 7 25 25 25 25 3 3 3 3 7 17 17 10 7 10 17 10 20 7 20 20 10 28 45 51 46 40 45 46 51 45 31 28 31 58 34 34 28 28 70 34 31 17 31 65 58 68 55 58 65 68 46 68 55 55 65 46 40 40 51 80 45 40 20 34 85 80 75 70 75 80 85 51 70 85 75 68 70 75 80 65 85 55 58 6 5 5 9 10 16 19 24 27 33 34 39 44 46 53 57 58 64 67 70 77 79 84 9 24 24 24 24 5 5 5 5 9 16 16 10 9 10 16 10 19 9 19 19 10 27 44 53 46 39 44 46 53 44 33 27 33 58 34 34 27 27 70 34 33 16 33 64 58 67 57 58 64 67 46 67 57 57 64 46 39 39 53 79 44 39 19 34 84 79 77 70 77 79 84 53 70 84 77 67 70 77 79 64 84 57 58 6 6 1 14 15 16 17 30 31 32 33 42 43 44 45 66 67 68 69 70 71 72 73 14 30 30 30 30 1 14 14 14 1 15 15 16 1 16 15 16 17 1 17 17 15 42 45 43 44 31 43 44 45 43 32 31 32 67 33 33 31 31 70 33 32 16 66 67 68 69 32 67 68 69 44 69 66 66 68 44 42 42 45 72 43 42 17 70 72 73 71 33 71 72 73 45 70 73 71 69 70 71 72 68 73 66 67 6 7 4 9 11 14 20 23 28 30 37 41 42 48 51 56 59 65 66 70 77 79 9 23 23 23 23 4 4 4 4 9 14 14 11 9 11 14 11 20 9 20 20 11 28 42 51 48 41 42 48 51 42 30 28 30 59 37 37 28 28 70 37 30 14 30 65 59 66 56 59 65 66 48 66 56 56 65 48 41 41 51 79 42 41 20 37 84 79 77 70 77 79 84 51 70 84 77 66 70 77 79 65 84 56 59 6 8 3 8 13 14 19 25 27 30 36 40 42 47 53 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8 10 15 10 21 8 21 21 10 27 45 50 48 39 45 48 50 45 30 27 30 60 36 36 27 27 72 36 30 15 30 65 60 67 54 60 65 67 48 67 54 54 65 48 39 39 50 79 45 39 21 36 85 79 74 72 74 79 85 50 72 85 74 67 72 74 79 65 85 54 60 8 5 3 7 10 17 20 24 29 30 35 41 44 47 50 54 59 64 69 71 74 81 84 7 24 24 24 24 3 3 3 3 7 17 17 10 7 10 17 10 20 7 20 20 10 29 44 50 47 41 44 47 50 44 30 29 30 59 35 35 29 29 71 35 30 17 30 64 59 69 54 59 64 69 47 69 54 54 64 47 41 41 50 81 44 41 20 35 84 81 74 71 74 81 84 50 71 84 74 69 71 74 81 64 84 54 59 2011) 2( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد الكامل في الفضاء االسقاطي ثالثي االبعاد على حقل كالوا (k,r)الغطاء GF(4) فاطمة فیصل كریم جامعة بغداد،ابن الهیثم -كلیة التربیة،قسم الریاضیات 2010، حزیران، 27: استلم البحث في 2010،كانون االول، 12: قبل البحث في الخالصة PG(3,pثالثـة ابعــاد فـي الفضــاء االسـقاطي ذي (k,r)الهـدف مـن هــذا البحـث هـو بنــاء الغطـاء حـول حقـل كــالوا ( GF(4) . وقد وجدنا ان اعظم غطاء كامـل(k,2) الـذي یـدعى اهلیلجـي، موجـود فـيPG(3,4) عنـدماk=13 .فضـال عـن .(k,5)و (k,4)و (k,3)ذلك وجدنا اعظم غطاء لـ .الفضاء االسقاطي أعظم غطاء كامل حقل كالوا :الكلمات المفتاحیة