On cyclic of Steiner system (v); V=2,3,5,7,11,13

Transcription

1 On cyclc of Stener system (v); V=,3,5,7,,3 Prof. Dr. Adl M. Ahmed Rana A. Ibraham Abstract: A stener system can be defned by the trple S(t,k,v), where every block B, (=,,,b) contans exactly K-elementes taken from aset V- elements; every two dstnct block B and Bj have at most (t-) elements n common of V, the set V usually called the base set. Stener system can be represented geometrcal by lettng the varetes be ponts and representng a block by lne (not necessary straght ) through the ponts t contans. In ths paper we connected between Stener system and geometry where we ll fnd all blocks of S(, p+,p + p+ ); p s prme number usng cyclc Stener system puttng S( p+,p +p+)n one cycle. Introducton: A specal knd of bloke desgns s called a Stener system. The dea a rose n November (85) by Stener who posed a many questons concernng what so called now a Stener trple systems. The second of these questons asked whether or not t was always possble to ntroduce a Stener trple system on V-ponts. Stener system: [3] Gven three ntegers t, k and V ; t< k < V a Stener system S(t, k,v) s a set V whch contans V ponts ( or vertes ) together wth a famly B of k-subset of V s contaned n exactly one block. Equvalently an S(t,k,V) s a t-(v,k,i) desgn.

2 Lemma: [6] Let S be a stener system wth the parameters (t,k, V) and let I V be an -set ; 0 t. then = where s the number of blocks contanng and depends only on the parameters V,k,t and, whle t s ndependent on. Lemma: [3] A S(t, k, V) exsts only f, s an nteger for every =0,,..,t-. Lemma: (Fsher s Inequalty): [] Let S be a Stener system wth the parameters (,k,v), then b V,where b s the number of blocks n S. Block desgn: A block desgn s an arrangement of V dstnct objects nto blocks ; each block contans exactly K dstnct objects, each object occurs n exactly r dfferent blocks. Furthermore every par of dstnct object a,a j occurs n exactly blocks. Galos feld: [5] Gven a postve prme nteger p, let k be a set wth P elements {0,,,,p-}.Defne addton n k by a+b= c f c s the remander of a+b devded by p,.e. a+b=c f c s a+b reduced modulo P. smlary

3 multplcaton n K s defned by a.b=c f c s the remander of a.b on dvdng by p. If m s a postve nteger and a k then we denoted by ma the element of K obtaned by addng a to tself m tmes n k,.e ma=b f b s ma reduced modulo p. Smlarly a m =b f b s a m reduced modulo p. Then k wth the two operatons, addton and multplcaton, defned as above s a feld wth zero and multplcaton dentty l whch s usually called Galos feld wth characterstc p and denoted by GF(p). For every a GF(p), pa=0 and a p =0 n partcular, -a=(p-)a, and f a 0 then a = a ( p ) Projectve plane : [] A projectve plan s a trple (P,L, I) where P s a set of objects(called ponts), L s a certan subsets of P (called lnes), and I s a relaton between the ponts and lnes; () every par of lnes ntersect n exactly one pont, () every par of ponts les n exactly one lne, (3) there exst four dstnct ponts, no three of them are contened n one lne. Theorem (): [5] The ponts (x, x, x 3 ) ncdence on the lne [y, y, y 3 ] f and only f x y +x y +x 3 y 3 =0 Theorem (): [5] If{p, p,p ( n ) } and {p,p,,p 3 ( n ) }are any two sets of n+ ponts of PG(n,k); n+ ponts are chosen from same set le n a prme then there exsts a unque projectvty A; P =P.A for all n N n

4 Theorem(3) : [3] Let S be asymmetrc stener system wth the parameters (,k,v).then S s a projectve plane wth parameters (,p+,p +p+). Cyclc stener trple system: [] An automorphsm of a STS(S,T) s abjecton α:s S such that T={x, y, z} T f and only f t ={x,y,z } T, ASTS(V ) s a cyclc f t has an automophsm that s a permutaton consstng of a sngle cycle of length V. Group acton on a set: [4] Def: Let X be a set and G a group. An acton of G on X s a map: *: X G such that: () x.e= x, for all x X, () x(g.g )=(xg )g,for all x X and all g,g G Under these condtons, X s a G-set. Theorem: [4] Let X be a G-set. For x,x X, Let x ~ x f and only f there exsts g G such that : x g = x. Then ~ s an equvalence relaton on S. Def: [4] Let X be a G-set. Each cell n the partton of the equvalence relaton descrbed n the theorem above s an orbt n X under G. 4

5 If x X, the cell contanng x s the orbt of x. We let ths cell be G x. Stener system S(,3,7) as PG(,): The projectve plane K= PG(,) contans 7 ponts and lnes, 3 ponts n every lne and 3 lnes through every pont. Snce (k,+) s abelan group wth dentty O, and (k/{0},*) s abelan group wth dentty,and dstrbutve law s satsfed, then (k,+,*) s a feld. Let P and L ; =,,,7 be the ponts and the lnes of PG(,) respectvely, and let A be a square matrx (cyclc projectvty); A = Then P =P A are the 7 ponts of PG(,).Wrtng for P, the vectors of The 7 ponts of PG(,) are gven n table () Table (). The ponts of PG (,) P (,0,0) (0,,0) 3 (0,0,) 4 (,,0) 5 (0,,) 6 (,,) 7 (,0,) By theorem (), the ponts(,0,0),(0,,0) and(,,0) ncdence on the lne L =[0,0,], L s the frst lne whch contans the ponts, and 4 then L = L A, =,,,7 are the lnes of PG(,). 5

6 Projectve plane PG(,) can be wrtten as S(,3,7) and ths set satsfy the condton of stener system where, = 7, whch s nteger, = 3, s an nteger too. So, the set S(,3,7) s realy stener system, the ponts of PG(,) are the ponts of stener system and the lnes of PG(,) are the blocks of stener system and the number of lnes equvalent to the number of blocks below. All block of S(,3,7) are gven n table (). Table(). The block of S(,3,7) Stener system S(,4,3) as PG(,3) : The projectve plane K=PG (,3) contans 3 ponts and 3 lnes, 4 ponts n every lne and 4 lne through every pont. (K,+) s abelan group wth dentty O, and (K/{0},*) s abelan group Wth dentty, and the dstrbutve law s satsfed, then (K,+,*) s a feld. 6

7 Gven cyclc projectvty, A = on PG(,3) therefor, the 3 ponts on PG(,3) are gven n table (3) below : Table (3) ponts of PG (,3) P (,0,0) (0,,0) 3 (0,0,) 4 (,,0) 5 (0,,) 6 (,,) 7 (,,) 8 (,,) 9 (,0,) 0 (,,0) (0,,) (,,) 3 (,0,) Let L be the lne whch contans the ponts,,4 and 0, then L = L A,=,, 3 are the lnes of PG(,3). Projectve plane PG(,3) can be wrtten as S(,4,3) and satsfes the condtons of stener system where, = 3, whch s an nteger, 7

8 = 4, s an nteger. So, the set S(,4,3) s really stener system. All blocks of S(,4,3) n table (4) below. Table (4). The block of S (,4,3) Stener system S(,6,3) as PG(,5) : By the same way we can fnd the ponts of PG(,5) where the cyclc Projectvty A =, and all ponts of PG(,5) gven n table(5). 8

9 Table (5). The ponts of PG (,5) P P P (,0,0) (3,,) (3,,0) (0,,0) (,,) (0,3,) 3 (0,0,) 3 (3,,) 3 (4,0,) 4 (,0,) 4 (3,4,) 4 (,4,) 5 (,,) 5 (,,0) 5 (,,0) 6 (3,3,) 6 (0,,) 6 (0,,) 7 (4,,) 7 (,0,) 7 (3,0,) 8 (,3,) 8 (,,) 8 (,3,) 9 (4,3,) 9 (,,) 9 (4,4,) 0 (4,,) 0 (,4,) 30 (4,,0) 3 (0,4,) And table (6) we can fnd all blocs of S(,6,3)

10 Stener system S(,8,57) as PG(,7) : Usng the same way to construct S (,8,57) and the frst block s and the last block s Stener system S (,,33) as PG(,) : By the same way we can fnd all blocks of S(,,33) and the frst block s

11 and the last block s: Stener system S (,4,83) as PG (,3): Usng the same way to construct S (,4,83) and we can fnd all blocks and the frst block s: and the last block s:

12 References: - B.H. Gross, Intersecton trangles and block ntersecton number of Stener system, math. Z,39(974), C.C.Lnder,C.A.Roger,Desgn theory (prntng). 3- C.J. Colborn; J. H. Dntz, the CRC hand book of combnateral Desgns. (prntng) (998) 4- J. B. Fralegh, Frst course n Abstract algebra. Thrd edton. Department of math. Un Of Rhyde Island, J.W.P.Hrschfeld. Projectve Geometres over fnte feld. Oxford (979). 6- K. Hanrch,Lacture notes. (prntng) (998).