2) 3( 22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد 009 PG(2,q)أنواع جدیدة من المجموعات القالبیة في المستوي االسقاطي میساء جلیل محمد ، علي طالب محمد جامعة بغداد -ابن الهیثم -كلیة التربیة -قسم الریاضیات الخالصة لقـد تـم فـي هـذا البحـث الحصـول علـى انـواع جدیـدة مـن المجموعـات القالبیـة فـي المسـتوي االسـقاطي حـول حقـل كـالوا PG(2,q) .بعــض كــذلك تــم الحصــول علــى . ن همــا المجموعــة القالبیــة الكاملــة والمجموعــة القالبیـة العظمــىان النوعــاهـذ .المجموعات حولالنتائج IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 New Kinds of Blocking sets in a Projective Plane PG(2,q) A.T. Mahammad, M. J. Mahammad Department of Mathematics, College of Education I bn-Al-Haitham , Unive rsity of Baghdad Abstract In this work, new kinds of blocking sets in a p rojective p lane over Galois field PG(2,q) can be obtained. These kinds are called the complete blocking set and maximum blocking set. Some results can be obtained about them. Introduction Let PG(2,q) denotes the 2-dimensional p rojective p lane over Galois field GF(q). A blocking set in PG(2,q) is a set of p oints that nonemp ty intersection with every line in PG(2,q), (1). This p aper is divided into two sections, section one consists of st andard basic, theorems and definitions of p rojective p lane, blocking set, minimal, trivial, t-fold, maximal, comp lete, and committee blocking sets. Section two consists of the relation between (k,n)-arc and blocking sets. Section One De fini tion "Projective Plane" (1) A p rojective p lane PG(2,q) over Galois field GF(q) is a two-dimensional p rojective sp ace, which consists of p oints and lines with relation between them, in PG(2,q) there are q 2 + q + 1 p oints, and q 2 + q + 1 lines, every line contains 1 + q p oints and every p oint is on 1 + q lines, any p oint in PG(2,q) has the form of a trip le (a1,a2,a3) where a1, a2, a3  GF(q); such that (a1,a2,a3)  (0,0,0). Two p oints (a1,a2,a3) and (b1,b2,b3) represent the same point if there exists  GF(q)\{0}, such that (b1,b2,b3) =  (a1,a2,a3). Similarly any line in PG(2,q) has the form of a trip le [a1,a2,a3], where a1,a2,a3  GF(q); such that [a1,a2,a3] ≠ [0,0,0]. Two lines [a1,a2,a3] and [b1,b2,b3] represent the same line if there exists  GF(q)\{0}, such that [b1,b2,b3] =  [a1,a2,a3]. There exists one p oint of the form (1,0,0). There exists q p oints of the form (x,1,0). There exists q 2 p oints of the form (x,y ,1). A p oints p (x1,x2,x3) is incident with the line L[a1,a2,a3] if and only if a1x1 + a2x2 + a3x3 = 0, i.e. a point rep resented by (x1,x2,x3) and the line rep resented by 1 2 3 a a a           , then (x1,x2,x3) 1 2 3 a a a           = 0  a1x1 + a2x2 + a3x3 = 0. Any p rojective plane PG(2,q) satisfies the following axioms: 1. Any two dist inct lines intersected in a unique p oint. 2. Any two dist inct p oints are contained in a unique line. 3. There exists at least four p oints such that no t hree of them are collinear. De fini tion "Blocking S et"(1) A blocking set B of PG(2,q) is a set of p oints intersecting every line of PG(2,q) in at least one point, so B is blocking set if and only if PG(2,q)\B is blocking set. IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 De fini tion "Minimal Blocking S et" (1) A Blocking set B is called minimal in PG(2,q) when no p rop er subset of it is st ill a blocking set such that,  p  B, B\{p} is not a blocking set. Example In PG(2,3), there exists 13 p oints 1,2,…,13 and 13 lines L1, L2,…, Ln, table (1), such that every p oint is on four lines and every line contains four p oints, let B={1,2,4,5,6,7}. B is a minimal blocking set of PG(2,3). De fini tion "Trivial Blocking S et"(2) A blocking set in PG(2,q) is called trivial when contains a line. De fini tion "Commi tte Blocking S et"(2) A blocking set B of PG(2,q) is committee if and only if B contains a minimum value of humber of p oints. Now the definition of complete blocking set can be given De fini tion "Comple te Blocking S et" A Blocking set B of PG(2,q) is called comp lete blocking set if for every p in PG(2,q)\B, B{p } is not blocking set or B{p } is trivial blocking set. i.e. B is comp lete blocking set if  p  B,  L  PG(2,q) and p  L such that L  (B{p }) = L. B is incomplete blocking set of PG(2,q) if there exists at least one p oint p  PG(2,q) such that B{p } is still blocking set. Example In PG(2,q), there exists 13 p oints and 13 lines showed in the table (1), such that in the p rojective plane PG(2,3) in the above examp le. Let B1={1,2,3,4,5,6}. B1 is a blocking set of PG(2,3), since it intersects every line in at least one point. PG(2,3)\B1 = {7,8,9,10,11,12,13} is also blocking set. B1 is incomplete blockin g set, since 7  PG(2,3)\B1 and B1{7} is still blocking set. Let B2={1,2,3,4,5,6,7}. B2 is a blocking set of PG(2,3) Since L1  B2  {8}, L2  B2  {9}, L3  B2  {10}, L4  B2  {11}, L5  B2  {12}, L6  B2  {13}. T hen B2  {8}, B2  {9}, B2  {10}, B2  {11}, B2  {12} and B2  {13} are not blocking sets and hence B2 is a complete blocking set. De fini tion "Ne w" "Maximum Blocking S et" A blocking set B of PG(2,q) is maximum if and only if PG(2,q)\B is committee blocking set. Example In PG(2,5) there exists 31 p oints {1,2,…,31} and 31 lines {L1,L2,…,L31}, in table (2), such that every p oint is on six lines and every line contains six p oints. Let B1=(1,2,3,4,5,7,10,17,21,27}, is a committee blocking set, and Let B2 = PG(2,5)\B1={6,8,9,11,12,13,14,15,16,18, 19,20,22,23,24,25,26,28,29,30,31} is a blocking set and it is maximum blocking in PG(2,5). Since it does not exist a blocking set B in PG(2,5) such that B2  B. De fini tion "A (k,n)-arc" (2) A (k,n)-arc in PG(2,q) is a set S of k points with p rop erty that every line contains at most n p oints of S, a (k,n)-arc S is called comp lete arc if it is not contained in a (k+1,n)-arc. De fini tion "t-Fold Blocki ng S et" (2) IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 A t-fold blocking set B in PG(2,q) is a set of p oints in PG(2,q) that intersects every line in PG(2,q) in at least t p oints. The orem (1) A blocking set B is minimal if and only if for every p oint pB, there is some line L such that B  L= p. Proof: Let B satisfies the condition that p is a p oint such that B  L = {P} for some line L, then B\{p} does not intersect L and so is not a blocking set and that is a contradiction. Thus B is minimal. Conversely sup p ose that B is minimal, so B\{p} is not a blocking set i.e. for if the condition is not satisfied there is some p oint p in B such that all lines through p contain another p oint of B so B/{p} is a blocking set and B is not minimal p oint pB there exists some line L  PG(2,q) such that p = L  B. The orem (1) If B is a blocking set in PG(2,q), and B= b, then (i) No line of PG(2,q) contains more that b – q p oints of B, (ii) If a line contains n p oints of B, then b  n + q, (iii) There is some line containing at least three p oints of B, (iv) b  q + 3. De fini tion "S ubgeome try" Let PG(2,q), q is square be a p rojective p lane, then PG(2, q ) is a subgeometry of PG(2,q) and contains q + q + 1 p oints and lines, every line contain q + 1 p oint and every p oints is on q + 1 lines. The orem (1), (3) In PG(2,q), q is square, then (i) If B= q + q + 1, B is blocking set, then B is subgeometry of PG(2, q ), (ii) If B= q 2 – q , B is blocking set, then B is complement of a subgeometry PG(2, q ). The orem (1) In PG(2,q), q is square and let B= b, B is blocking set. Then q + q + 1  b  q2 – q . Proof Sup p ose b= q + q + 1 – n , n > 0. By theorem (1.14) no line in PG(2,q) contains more than q + 1 – n p oints of B. Let S be any set of n p oints such that S  B =  and S = S  B is not subgeomety of PG(2,q), then S is a blocking set since no line contains more than ( q + 1 – n) + n = q + 1 of its p oint. So by theorem (1.16) S is a subgeomety , contradicting the choice of S. If B > q 2 – q , then  \ B < q + q + 1, where  = PG(2,q). The orem Let B be a blocking set in PG(2,q), then B is a minimal blocking set if and only if B* is comp lete blocking set. (B* = PG(2,q)\B) IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Proof: Sup p ose that B is minimal blocking set, then B* is blocking set. (by definition 1.2). Now, we must p rove B* is complete sup p ose that B* is not comp lete blocking set. Then there exists p  B* such that B*{p } is blocking set. Hence PG(2,q) \(B*{P}) is blocking set (by definition 1.2), but PG(2,q) \(B*{P}) = B\{p } t hat contradiction, since B is minimal, then B* is complete blocking set. Conversely , sup p ose that B* is a complete blocking set, to p rove that B is a minimal blocking set. Sup p ose that B is not minimal, then there exists P  B such that B\{p} is blocking set, then PG(2,q)\(B\{p }) is a blocking set (by definition 1.2) but PG(2,q)\(B\{p }) = B*{p }, and that is a contradiction, since B* is a comp lete blocking set. Then B is a minimal blocking set. Section Two Blocking S ets and (k,n)-arcs (4) A (k,n) –arc in PG(2,q) is a set S of k p oints with p rop erty that every line contains at most n p oints of S, most authers add the condition, that there should be some line meeting S in exactly n p oints, there is an obvious relation between (k,n)-arcs and blocking sets, the comp lement of (k,n)-arc is a (q – n)-fold blocking set of size q. The orem (5) In PG(2,q). A (k,n)-arc K is comp lete if and only if B=PG(2,q)\K is (q + 1 – n)-fold minimal blocking set. Proof: sup p ose that K is a comp lete (k,n)-arc, it is clear that B is (q + 1 – n)-fold blocking set and B= q 2 + q + 1 – k, to p rove that B is minimal blocking set, sup p ose B is not minimal, then there exists p  B such that B\{p} is blocking set, but B is (q + 1 – n)-fold blocking set and B= q 2 + q + 1 – k, t hen K  {p } is a (k+1,n)-arc, and k  (k+1,n)-arc (contradiction), since K is complete, then B is minimal. Conversely sup p ose that B is (q + 1 – n)-fold minimal blocking set and B=q 2 + q +1– k, it is clear that K is (k,n)-arc, to p rove K is comp lete sup p ose that K is not comp lete arc, then there exists p  K such that K  {p } is a (k+1,n)-arc but PG(2q)\K{p }= B\{p} is blocking set and that is contradiction, since B is minimal blocking, then K is comp lete arc. Open Problem (5) In Simeon Ball (6), he gives an op en p roblem which say s that, there exists a t-fold blocking set of PG(2,q) of size less than (t + 1) p ". And he found the answer is no if q = 3, 5 and 7. And we check this p roblem if q = 4, 5, 8. 1. In PG(2,4), there is no 2-fold blocking set of size less than 12 p oints, and the all 2-fold blocking set we found exactly 12 p oints, for examp le, B={1,2,4,5,6,7,9,10,12,13,17,19}, and the all 2-fold blocking set of size less than 12 p oints that we found it is trivial blocking set. 2. In PG(2,5), there is no 3-fold blocking set of size less t han 20 p oints, and the only 3-fold blocking set that we found tis exactly size is 20 p oints, for examp le B={1,2,3,5,6,7,8,9,11,12,14,16,20,24,27,28,29,30,31}, in PG(2,5), the all 3-fold blocking set of size is less t han 20 points that we found which is trivial blocking set. 3. In PG(2,8), there is no 3-fold blocking set of size less than 33 p oints, for examp le B={1,2,3,5,7,10,11,12,13,15,16,17,19,20,27,29,33,34,35,37,45,46,47, 50,56,58,60,61,67,69,70,72,73}, and the all 3-fold blocking set of size 33 p oints that we found which is t rivial blocking set. Re ferences 1. Hirschfeld, J.W.P. (1979). Projective Geometries Over Finite Field, Oxford University . 2. Hirschfeld, J.W.P. and Storm, L., (1998), Projective Geometry , Oxford University p ress, Oxford. IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 3. Art Blokhuis, Storm, L. and Szony i, T. (1999). J. London M ath. Soc. (2):321-332 .رسالة ماجستیر، جامعة الموصل ،(k,n)-arc ، القید األعلى لالقواس)2001(ن عبد الكریم، با .4 5. Simeon Ball, (2004), "Affine Blocking Sets", University of Politenicade Catauny a (Barcelonal). Table (1)The Points and Lines of PG(2,3) i Pi Li 1 (1,0,0) 1 2 4 10 2 (0,1,0) 2 3 5 11 3 (1,1,0) 3 4 6 12 4 (2,1,0) 4 5 7 13 5 (0,0,1) 5 6 8 1 6 (1,0 ,1) 6 7 9 2 7 (2,0,1) 7 8 10 3 8 (0,1,1) 8 9 11 4 9 (1,1,1) 9 10 12 5 10 (2,1,1) 10 11 13 6 11 (0,2,1) 11 12 1 7 12 (1,2,1) 12 13 2 8 13 (2,2,1) 13 1 3 9 IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Table (2)The Points and Lines of PG(2,5) i Pi Li 1 1 0 0 2 7 12 17 22 27 2 0 1 0 1 7 8 9 10 11 3 1 1 0 6 7 16 20 24 28 4 2 1 0 4 7 14 21 23 30 5 3 1 0 5 7 15 18 26 29 6 4 1 0 3 7 13 19 25 31 7 0 0 1 1 2 3 4 5 6 8 1 0 1 2 11 16 21 26 31 9 2 0 1 2 9 14 19 24 29 10 3 0 1 2 10 15 20 25 30 11 4 0 1 2 8 13 18 23 28 12 0 1 1 1 27 28 29 30 31 13 1 1 1 6 11 15 19 23 27 14 2 1 1 4 9 16 18 25 27 15 3 1 1 5 10 13 21 24 27 16 4 1 1 3 8 14 20 26 27 17 0 2 1 1 17 18 19 20 21 18 1 2 1 5 11 14 17 25 28 19 2 2 1 6 9 13 17 26 30 20 3 2 1 3 10 16 17 23 29 21 4 2 1 4 8 15 17 24 31 22 0 3 1 1 22 23 24 25 26 23 1 3 1 4 11 13 20 22 29 24 2 3 1 3 9 15 21 22 28 25 3 3 1 6 10 14 18 22 31 26 4 3 1 5 8 16 19 22 30 27 0 4 1 1 12 13 14 15 16 28 1 4 1 3 11 12 18 24 30 29 2 4 1 5 9 12 20 23 31 30 3 4 1 4 10 12 19 26 28 31 4 4 1 6 8 12 21 25 29