Journal of Babylon University/Pure and Applied Sciences/ No.(4)/ Vol.(20): 2012 ١١٣٨ Existence of Minimal Blocking Sets of Size 31 in the Projective Plane PG(2,17) Nada Yassen Kasm Yahya Department of Mathematics, College of Education University of Mosul Abstract In this paper, we show that by an example existence minimal blocking set of Rédei-type of size 27 representing protectively triangle in PG(2,17) example (2-5).We prove that the projective plane PG(2,17)having minimal blocking set of size 31 contain 14-secant and not contain i-secant ;14<i<q theorem (3-1) ,also we finding important propositions about minimal blocking set of size 31 in the projective plane PG(2,17)in theorems which are(3-3) into(3-19). ﺍﻟﺨﻼﺼﺔ ﺍﻟﺒﺤ ﻫﺫﺍ ﻓﻲ ﺙ ﻗﻤ ﻨﺎ ﺍﻟﻨﻭﻉ ﻤﻥ ﺃﺼﻐﺭﻴﻪ ﻗﺎﻟﺒﻴﻪ ﻤﺠﻤﻭﻋﺔ ﻭﺠﻭﺩ ﺒﻤﺜﺎل ﺒﺈﺜﺒﺎﺕ_ ﺤﺠﻡ ﺫﺍﺕ ﺭﻴﺩﻱ٢٧ ﺍﺴـﻘﺎﻁ ﻤﺜﻠﺜـﺎ ﻭﺘﻤﺜـل ﻓـﻲ ﻴﺎ PG(2,17) ﻤﺜﺎل) ٥ - ٢ ( . ﻭﺃﺜﺒﺘ ﻨﺎ ﺃﻹﺴﻘﺎﻁﻲ ﺍﻟﻤﺴﺘﻭﻱ ﺃﻥPG(2,17) ﺍﻟﺤﺠﻡ ﺫﺍﺕ ﺃﺼﻐﺭﻴﻪ ﻗﺎﻟﺒﻴﻪ ﻤﺠﻤﻭﻋﺔ ﻴﻤﺘﻠﻙ٣١ ﻗﺎﻁﻊ ﺘﻀﻡ ﻭﺍﻟﺘﻲ- ١٤ ﻗﺎﻁﻌﺎ ﺘﻀﻡ ﻭﻻ- i ; q < i < ١٤ ﻤﺒﺭﻫﻨﺔ) ١ - ٣ ( . ﺍﻟﺨﻭﺍﺹ ﻭﺠﺩﻨﺎ ﻭﺃﻴﻀﺎ ﺤﺠـﻡ ﺫﺍﺕ ﺍﻻﺼﻐﺭﻴﺔ ﺍﻟﻘﺎﻟﺒﻴﺔ ﺍﻟﻤﺠﺎﻤﻴﻊ ﺤﻭل ﺍﻟﻤﻬﻤﺔ٣١ ﺍﻻﺴﻘﺎﻁﻲ ﺍﻟﻤﺴﺘﻭﻱ ﻓﻲPG(2,17) ﺍﻟﻤﺒﺭﻫﻨﺎ ﻓﻲ ﺕ ﻤﻥ ﻫﻲ ﺍﻟﺘﻲ) ٣ - ٣ ( ﺍﻟﻰ) ١٩ - ٣ .( 1. Introduction Let GF(q)be denote the Galois field of q elements and V(3, q) be the vector space of row vectors of length three with entries in GF(q). Let PG(2, q) be the corresponding projective plane. The points of PG(2, q) are the non-zero vectors of V(3, q) with the rule that X = (x 1 ; x 2 ; x 3 ) and Y = (λx 1 ; λx 2 ; λx 3 ) are the same point, where λ ∈ GF(q)\{0}. Since any non-zero vector has precisely q-1non- zero scalar multiples, the number of points of PG(2, q) is 3 1 1 q q − − = q 2 +q+1. If the point P(X) is the equivalence class of the vector X, then we will say that X is a vector representing P(X). A subspace of dimension one is a set of points all of whose representing vectors form a subspace of dimension two of V(3, q). Such subspaces are called lines. The number of lines in PG(2, q) is q 2 + q + 1. There are q + 1 points on every line and q + 1 lines through every point. Also ,if V is vectors spaces of dimension two define on the field GF(q). then any subset from V which are meet for all prime from V counting at least n(q-1)+1points[ Hirschfeld, J.W.P. (1979)]. A blocking set in a projective plane is a set B of points, such that every line contains at least one point of B. If B contains a line, it is called trivial. If no proper subset of B is a blocking set it is called minimal [ Hirschfeld, J.W.P. and Storme, L. update(2001)] .Let B be a non-trivial minimal blocking set, and let be a line containing l < q + 1 points of B. Then it follows immediately that |B| ≥ q + l, by considering the lines through a point P of L not belonging to the blocking set. If we have equality, then every line through P different from L contains precisely one point of B .Blocking sets of this kind are called of Rédei- type and were studied in [ Bruen,A.A. and Thas,J.A. (1977)] and in [ Blokhuis,A. A.and Brouwer, E.and . Sz˝onyi, T. (1995)].We call B of Rédei- type if there exists a line l such that |B \ l| = q (the line l is called a Rédei line of B. [ Dipaola,J.(1969)]made idea about projective triangle which are an example of a blocking set of size 3(q+1)/2 in the Desarguesian planes of odd orders. The Elation α in the projective plane PG(2,q) is bijection fixed the points of l , and reverse the lines passing through p on l [Innamorate and Storme, 2004]. (1-1)Theorem:[ Barát ,.and Innamorati ,2003]