Basrah Journal of Science (A) Vol.34(3), , 2016

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1 Basrah Joural of Sciece (A) Vol.34(3), , 216 Miimax arc of kid (, δ ) with weighted poits Mohammed Yousif Abass Basrah Uiversity, College of Sciece, Mathematics Departmet E mail: mohammedyousif42@yahoo.com Abstract I this paper we proved that the existece of ( k, s; f ) arcs of type ( r, s ) i PG( 2, q ); wheq =, derived from a miimax arc Cof kid (, δ ), that is a (( 1)q + 1, ) arc. We fid these ( k, s; f ) arcswhes r = q, as i the table below. We deote by MIN for miimal, MAX for maximal ad X for either miimal or maximal. Summary for = (( 1)q + 1, ) arc. [ q =, > 2 ] Weights 1 1 t 2 of E t 1 1 poits I t W MIN X MAX X MAX MIN X Keywords:projective plae of order q, miimax arcs, (k, ; f) arc of type (m, ), arcs with weighted poits. Itroductio I 1991, the author B. J. Wilso (1) discussed the existece of the miimax arcs i PG(2, q), whe q is a odd prime power. We use the results of (1) ad combiig them with the results of (2, 3, 4, 5, 6, 7, 8) to obtai the miimax arc with weighted poits. Let π be a projective plae of order q, ad deote by P ad R respectively the set of poits ad lies of π. Let f be a fuctio from P ito the set N of o egative itegers, ad call the weight of p P, the value f(p ), ad the support of fthe set of poits of the plae havig o zero weight. Usig f we ca defie the fuctio N F: R such that for ay r Rwe have F(r) = f(p ) p r. We call F(r) the weight of the lie r. Defiitio 2. (3) Defiitio 1. (2) 137

2 Mohammed Yousif Abass Miimax arc of kid (, δ ) with A (k, ; f) arc K i PG(2, q) is a fuctio f P N such that k = The support of f ad = max F. Observe that a (k, ) arck is a (k, ; f ) arc if f is chose by:, ifp K f( p ) = { 1, ifp K. Defiitio3. (3) The type of a (k, ; f) arck is the set of Im F. To write explicitly the type of K we ca use the sequece ( 1,, ρ ) where λ Im F, λ = 1,, ρ ad 1 < 2 < < ρ =. Notatio (i) ω = max f. (ii) [p] deote the set of lies passig through p. Lemma 1. (4) Let K be a (k, ; f ) arc i PG(2, q). The for every poit p we have F(r) = W + qf( p ). r [p] Lemma 2. (4) The weight W of a (k, ; f ) arc of type (m, ) satisfies: m(q + 1) W ( ω)q +. Defiitio 4. (4) We call arcs for which the values i lemma (2) are attaied, maximal ad miimal(k, ; f ) arcs of type (m, ) respectively. Theorem 1. (4) A ecessary coditio for the existece of a (k, ; f) arck of type (m, ), m > is that: (a) q mod( m); (b) ω m ; (c) m 2. Miimax arcs I PG(2, q), where q = is a odd prime power ad > 2, let C deote a (( 1)q + 1, ) arc. Let P C ad suppose that through P there pass ρ i i secat of C for i = 1, 2,,. The coutig lies through P gives ρ 1 + ρ ρ = q + 1 (1) Ad coutig poits of C o lies through P gives ρ 2 + 2ρ ( 1)ρ = ( 1)q.(2) Elimiatig ρ gives 138

3 Basrah Joural of Sciece (A) Vol.34(3), , 216 ( 1)ρ 1 + ( 2)ρ ρ 1 = 1.(3) The maximum value of ρ occurs whe ρ 1 = 1, ρ 2 = = ρ 1 =, ρ = q. Ad the miimum value of ρ occurs whe ρ 1 = ρ 2 = = ρ 2 =, ρ 1 = 1, ρ = q + 2. Defiitio 5. (1) If every poit of C is of oe of the above two types the C is called a mimax arc. The poits of C are called maximum or miimum poits accordigly. Suppose ow that C is a mimax arc havig μ maximum poits ad δ miimum poits. We shall refer to C as a miimax arc of kid (μ, δ ) whe appropriate. The, usig the otatio τ i to deote the umber of i secats of C we have μ + δ = ( 1)q + 1(4) τ 1 = μ(5) τ 1 = δ(6) τ = qμ+(q +2)δ q + μ q2 +q+2μ 2 (7) = q 2 + 3q 1 τ = (q 2 + q + 1) τ 1 τ 1 τ = (1 q μ) + q2 +q+2μ 2 (8) Let Q be a poit ot o C ad suppose that through Q there pass σ i i secats for i =, 1, 1,. The σ + σ 1 + σ 1 + σ = q + 1.(9)ad σ 1 + ( 1)σ 1 + σ = q q + 1.(1) From these we may obtai σ + ( 1)σ 1 + σ 1 = + q 1. (11) Now, suppose that C be of kid(,δ ) i PG( 2, q ) with q =. Through each poit of Care q + 2, secatad 1,( 1) secats. Hece the equatios (4) (8), become: δ = ( 1)q + 1 = ( 1)t +.(12) τ 1 =. (13) τ 1 = δ = ( 1)t +. (14) (q + 2)δ τ = (q + 2)(( 1)t + ) = = (q + 2)(( 1)) = (t + 3)(( 1)),(15) 139

4 Mohammed Yousif Abass Miimax arc of kid (, δ ) with τ = (1 q) + q2 +q 2 (t+1) 2 + t 1 (16) = t = t 2 + 3t t. Sice C be of kid (, δ ), the the equatios (9), (1) ad (11) becomes: σ + σ 1 + σ = t + 2. (17) ( 1)σ 1 + σ = ( 1)t +. (18) σ + σ 1 = + t. (19) From equatio (18) ad (19) we must have σ 1.Let l be a ( 1) secat. l has to meet (τ 1 1 = ( 1)q) other, ( 1) secat. Sice l cotai ( 1) poits of C ad through each poit there pass ( 2) lies differ from l, the it meets ( 1)( 2)lies o Cad ( 1)(q + 2) i its remaiig i ( 1) oes (q + 2) poits. Hece σ 1 = o l \C. Thus the poits ot o Cmay be classified as: Table (1) Sice C = ( 1)q + 1 = 2 t t +. E + I = q 2 + 2q q = 2 t 2 2 t + 4t + 3. From equatio (12.9) of Lemma (12.1)of (9), we have: (q + 1 i)τ i = σ i Q PG(2,q)\C (q + 2)τ 1 = σ 1. Q PG(2,q)\C (q + 2)(( 1)t + ) = σ 1 = E. Q E E = (t + 3)(( 1)t + ). E = (t + 3)(( 1)) = 2 t 2 t 2 2 t + 5t 3t + 3. The I = t 2 t + 3t. A secat meets ( 1),( 1) secats o C ad hece has to meet (τ 1 ( 1)) = (( 1)t + 2), elsewhere i 's. The o a Types σ secat are (( 1)t + 2) E σ 1 σ I poits ad hece q + 1 ( 1) E t (( 1)t + 2) = ( t, I 1)t poits. + O 2 a ( 1) secat are q + 2 = t + 3, E poits. A secat C + E + I = q 2 + q + 1 = 2 t 2 + 3t + 3. has to meet ( 1)q + 1, ( 1) secats i 's ad hece o it are 14

5 Basrah Joural of Sciece (A) Vol.34(3), , 216 ( 1)q+1 = ( 1)t+ = ( 1)t + 1,E poits ad hece q + 1 ( 1)q+1 = q+ 1 =, I poits. Hece we obtai the followig table: Table (2) F(( 1) secat) F( secat) = (t + 3) β. = (( 1)t + 1) β Lies Poits o C Poits o E + ()α. Poits o I secat (( 1)t + 2) t (i) Let F( secat) = F(( ( 1) secat 1 t + 3 1) secat). The secat ( 1) (( 1)t + 2) β + t α tα = ()β. = (t + 3) β. Miimax arc with weighted poits I this sectio, we discuss (k, ; f ) arc of type (r, s), Im( f ) ={, α, β} where α ad β are determied accordig to the weights of the lies ad Table (2). We will divide this sectio ito three cases ad each case partitio ito three subcases: Case (1): Now, i this case assig weights α to I poits, to poits of C ad β to E poits. Weights of lies are Setα =, β = t. F( secat) = (( 1)t + 2) t + t () = t 2 t + 3t. F(( 1) secat) = (t + 3) t = t 2 t + 3t. F( secat) = () 2 + (( 1)) t = t F( secat) = (( 1)t + 2) β + t α. r = t 2 t + 3t, s = t

6 Mohammed Yousif Abass Miimax arc of kid (, δ ) with Clearly that (s r) q, because s r = = q. Oly r weightig lies pass through poits of C= poits of weight. The we expect this arc to be miimal with ω = α = ad W = r(q + 1) = (t 2 t + 3t)(q + 1) = (t 2 t + 3t)( t + 2) = 2 t 3 + 5t 2 2 t 2 2t + 6t. Result 1. There exists (q 2 + 2q q, t 2 + 3; f ) arc of type (t 2 t + 3t, t 2 + 3), whe W is miimal ad the poits of weight are the poits of C. (ii)let F( secat) = F( secat). The (( 1)t + 2) β + t α = (( 1)) β + ()α. ( 1)β + α =, which is impossible. Because α >, β > ad > 2. (iii)let F(( 1) secat) = F( secat). (t + 3) β = (( 1)) β + ()α. (t + 2) β = ()α Set α = t + 2, β =, provides that < t + 2 t > 2. F( secat) = (( 1)t + 2)() + t (t + 2) = t 2 t + 3t + 2. F(( 1) secat) = (t + 3)() = t 2 + 3t + 3. F( secat) = (( 1))(t + 1) + ()(t + 2) = t 2 + 3t + 3. r = t 2 t + 3t + 2, s = t 2 + 3t + 3. Also, we have s r = = q. This arc is either maximal or miimal. Sice F(l) = W + q f(p). l [P] 142

7 Basrah Joural of Sciece (A) Vol.34(3), , 216 If P C f(p) =. W = F(l) l [P] W = (q + 2)F( secat) + ( 1)F(( 1) secat) = 2 t 3 2 t 2 + 5t 2 t + 6t + 3. Result 2. There exists (q 2 + 2q q, t 2 + 3t + 3 ; f ) arc of type ( t 2 t + 3t + 2, t 2 + 3t + 3), whe W is either miimal or maximal ad the poits of weight are the poits of C. Case (2): Assig weights α to I poits β to poits of C ad to E poits. Weights of lies are F( secat) = tα + β F(( 1) secat) = ( 1)β F( secat) = ()α. (i) Let F( secat) = F(( 1) secat) tα + β = ( 1)β tα = β, which is impossible. (ii)let F( secat) = F( secat). The tα + β = ()α β = α. Set α =, β = 1. F( secat) = t + = (). F(( 1) secat) = 1. F( secat) = (). r = 1, s = (). Sice through every poit of maximum weight ( α = )there pass olys weightig lies, the this arc is maximal. The W = (s α)q + s = tq + () = t() + () = 2 t 2 + 2t +. Result 3. There exists ( t t 2t + 3t +, (); f ) arc of type ( 1, ()), whe W is maximal ad the poits of weight are the poits of E. (iii) Let F(( 1) secat) = F( secat). The ( 1)β = ()α Set α = 1, β =, provides that t + 2. F( secat) = t( 1) + () = 2t t + = s. 143

8 Mohammed Yousif Abass Miimax arc of kid (, δ ) with F(( 1) secat) = ( 1)() = t t + 1 = r. F( secat) = ( 1)() = t t + 1 = r. s = 2t t +, r = t t + 1. This arc is either maximal or miimal. The F(l) = W + q f(p). l [P] If P E f(p) = W = F(l) l [P] W = (( 1)t + 2)F( secat) + F(( 1) secat) + t F( secat) = 2 2 t 2 2t 2 + 4t 3t +. Result 4. There exists ( t t 2t + 3t +, 2t t + ; f ) arc of type ( t t + 1, 2t t + ), whe W is either maximal or miimal ad the poits of weight are the poits of E. Case (3): Assig weights to I poits, α to poits of C ad β to E poits. Weights of lies are F( secat) = α + (( 1)t + 2)β. F(( 1) secat) = ( 1)α + (t + 3) β. F( secat) = (( 1))β. (i) Let F( secat) = F(( 1) secat). The α + (( 1)t + 2)β = ( 1)α + (t + 3) β. α = ()β. Set α =, β = 1. F( secat) = () + (( 1)t + 2) = (2 1)t + 2. F(( 1) secat) = ( 1)() + t + 3 = (2 1)t + 2. F( secat) = ( 1). 144

9 Basrah Joural of Sciece (A) Vol.34(3), , 216 s = (2 1)t + 2, r = ( 1), ω =. This arc is maximal. W = (q + 1)(s ω) + ω = (q + 1)((2 2)t + 1) + () = (t + 2)((2 2)) + () = (2 2)t 2 + (5 3)t + 3. Result 5. There exists ( 2 t 2 t 2 + 4t 3t + 3, (2 1)t + 2; f ) arc of type (( 1), (2 1)t + 2), whe Im( f ) = {, 1, }, W is maximal ad the poits of weight are the poits of I. (ii) Let F( secat) = F( secat). The α + (( 1)t + 2)β = (( 1))β. α = ( 1)β. Set α = 1, β =. F( secat) = ( 1) + (( 1)t + 2) = ( 1)t +. F(( 1) secat) = ( 1) 2 + (t + 3) = 2 t F( secat) = (( 1)) = ( 1)t +. s = 2 t + + 1, r = ( 1)t +, ω =. This arc is miimal. W = r (q + 1) = (( 1)t + )(t + 2) = 2 ( 1)t 2 + (3 2 2)t + 2. Result 6. There exists ( 2 t 2 t 2 + 4t 3t + 3, 2 t + + 1; f ) arc of type ( ( 1)t +, 2 t ), whe Im( f ) = {, 1, }, W is miimal ad the poits of weight are the poits of I. (iii) Let F(( 1) secat) = F( secat). The ( 1)α + (t + 3) β = (( 1))β. ( 1)α = ( (t + 2))β, which is impossible if (t + 2). 145

10 Mohammed Yousif Abass Miimax arc of kid (, δ ) with Now, we discuss the case i which > (t + 2). The we suppose that = m + (t + 2), m Z +. Set α = (t + 2) = m, β = 1 = m +. F( secat) = m + (( 1)t + 2)(m + ) = ( 1) 2 t t + 2. F(( 1) secat) = ( 1)m + (t + 3)(m + t + 1) = ( 1) 2 t + 1. F( secat) = (( 1))( 1) = ( 1) 2 t + 1. Sice ( 1) 2 t + 1 ( 1) 2 t + t + 2 = = q, the s = ( 1) 2 t + 1, r = ( 1) 2 t t + 2. Also this arc is either miimal or maximal. The we have F(l) = W + q f(p). l [P] If P I f(p) = W = F(l) l [P] W = (( 1))F( secat) + () F( secat) W = (( 1))(( 1) 2 t t + 2) + ()(( 1) 2 t + 1) = ( )t 2 + ( )t Result 7. There exists ( 2 t 2 t 2 + 4t 3t + 3, ( 1) 2 t + 1; f ) arc of type (( 1) 2 t t + 2, ( 1) 2 t + 1), whe Im( f ) = {, m, 1}, where m = (t + 2) Z +. Also W is either miimal or maximal ad the poits of weight are the poits of I. Refereces (1) B. J. Wilso, (1991), "Miimax arcs", Discrete mathematics 92, (2) M. Y. Abass, (211), "Existece ad o existece of (k, ; f ) arcs of type ( 3, ) i PG(2, 9)", 146

11 Basrah Joural of Sciece (A) Vol.34(3), , 216 M. Sc. Thesis, Uiversity of Basrah, Iraq. (3) E. D'Agostii, (1994), "Oweightedarcswiththreeo zerocharacters", Joural of Geometry, Vol. 5, (4) F. K. Hameed, (1989), "Weighted ( k, ) arcs i the projective plae of order ie", Ph.D. Thesis, Uiversity of Lodo Royal Holloway ad Bedford New College, Eglad. (5) F. K. Hameed,M. Hussei ad Mohammed Y. Abass, (211), "O (k, ; f ) arcs of type ( 5, ) i PG(2, 5)", Joural of Basrah Researches ((Scieces)) Volume 37. Number 4. C, (6) R. D. Mahmood, (199), "(k, ; f ) arcs of type ( 5, ) i PG( 2, 5 ) ", M.Sc. Thesis, Uiversity of Mosul, Iraq. (7) G. Raguso ad L. Rella, (1983), " Sui ( k, ; f ) architipo ( 1, ) di u piao proiettivofiito ", Note di Matematica Vol. III, (8) B. J. Wilso, (1986), " (k, ; f ) arcs ad caps i fiite projective spaces", Aals of Discrete mathematics 3, (9) J. W. P. Hirschfeld, (1998), "projective geometries over fiite fields", Secod Editio, Claredo Press, Oxford, 555+xiv pp. 147

12 Mohammed Yousif Abass Miimax arc of kid (, δ ) with القوس األعظم األصغر من النوع ) δ, )مع النقاط الموزونة محمد يوسف عباس قسم الرياضيات / كلية العلوم / جامعة البصرة البريد االلكتروني: mohammedyousif42@yahoo.com الخالصة في هذا البحث برهنا وجود األقواس - ) f ( k, s ; من النوع ) s ( r, في ) q, PG( 2, عندما = q (( 1)q + 1, ) - ( أي انه القوس, δ النوع( مشتقا من القوس األعظم األصغر Cمن,. نحن وجدنا األقواس - ) f ( k, s ; عندما r = q و كما مبين في الجدول أدناه. رمزنا بالرمز MAX للقيمة العظمى و MIN للقيمة الصغرى و X للتي ليست عظمى وال صغرى. Summary for = (( 1)q + 1, ) arc. [ q =, > 2 ] Weights 1 1 t 2 of E t 1 1 poits I t W MIN X MAX X MAX MIN X 148