Classification results on weighted minihypers Jan De Beule (Ghent - PDF document

Classification results on weighted minihypers Jan De Beule (Ghent University) joint work with: Leo Storme (Ghent University)

Optimal codes and related topics 2005 1 Introduction Consider ǫ t subspaces PG( t, q ) , ǫ t − 1 subspaces PG( t − 1 , q ) , . . . , ǫ 1 subspaces PG(1 , q ) (lines) and ǫ 0 points. The union of this objects is an t t � � { ǫ i v i ; n, q }− minihyper( F, w ) ǫ i v i +1 , i =0 i =1 v i = | PG( i, q ) | = q i +1 − 1 q − 1 Disjoint subspaces give a non-weighted minihyper. By a theorem of Hamada, Helleseth and Maekawa, also the reverse i =0 ǫ i � √ q . is true when � t question: can we prove the same result for weighted minihypers? Jan De Beule, Ghent University

Optimal codes and related topics 2005 2 The planar case The first step is the planar case. What exists for non-weighted minihypers in the plane? Non-weighted { f, m ; 2 , q } minihypers are often called m -fold blocking sets. Theorem 1. (S. Ball [1], K. Metsch) An ǫ 1 -fold blocking set in PG(2 , q ) , ǫ 1 small, not containing a line, has at least ǫ 1 q + √ ǫ 1 q + 1 points. it follows: Lemma 1. An { ǫ 1 ( q + 1) + ǫ 0 , ǫ 1 ; 2 , q } - minihyper ( F, w ) contains a line if ǫ 1 + ǫ 0 � √ q . Can we remove this line and still have a ( ǫ 1 − 1) -fold blocking set? Jan De Beule, Ghent University

Optimal codes and related topics 2005 3 The planar case (continued) Suppose that the { ǫ 1 ( q +1)+ ǫ 0 , ǫ 1 ; 2 , q } - minihyper ( F, w ) contains a line L . • If | ( F, w ) ∩ L | � q + ǫ 1 , reducing the weight of every point of L with one gives a new { ( ǫ 1 − 1)( q + 1) + ǫ 0 , ǫ 1 − 1; 2 , q } - minihyper ( F ′ , w ′ ) . • If q + 1 � | ( F, w ) ∩ L | � q + ǫ 1 − 1 , it is not immediately clear that this procedure works. It is possible that we have to add at most ǫ 1 − 1 points p i again. Jan De Beule, Ghent University

Optimal codes and related topics 2005 4 The planar case (continued) Using polynomial techniques we obtain that L is the only line on p i containing exactly ǫ 1 − 1 points of the new minihyper. But: Lemma 2. (A. Blokhuis, L. Storme and T. Sz˝ onyi [2]) Let ( F, w ) be an { ( ǫ 1 − 1)( q + 1) + c, ǫ 1 − 1; 2 , q } -minihyper, ǫ 1 − 1 + c < q 2 , and let p ∈ F be a point of weight 1 . Then p lies on at least q − c lines intersecting ( F, w ) in ǫ 1 − 1 points. This gives a contradiction, in other words: Theorem 2. An { ( ǫ 1 − 1)( q +1)+ ǫ 0 , ǫ 1 − 1; 2 , q } -minihyper, ǫ 1 + ǫ 0 � √ q , is a sum of ǫ 1 lines and ǫ 0 points. Jan De Beule, Ghent University

Optimal codes and related topics 2005 5 The situation in 3 -space We consider a { ( ǫ 1 − 1)( q + 1) + ǫ 0 , ǫ 1 − 1; 3 , q } -minihyper ( F, w ) , ǫ 1 + ǫ 0 � √ q . Projecting ( F, w ) from a point p �∈ ( F, w ) gives a { ( ǫ ′ 1 − 1)( q + 1) + ǫ ′ 0 , ǫ ′ 1 − 1; 2 , q } - minihyper ( F ′ , w ′ ) . Using that ( F ′ , w ′ ) is a sum of lines and points, we can prove that ( F, w ) is the sum of ǫ 1 lines and ǫ 0 points. Inductively, we can prove: Theorem 3. A { ( ǫ 1 − 1)( q + 1) + ǫ 0 , ǫ 1 − 1; k, q } -minihyper ( F, w ) , ǫ 1 + ǫ 0 � √ q , k � 2 , is a sum of ǫ 1 lines and ǫ 0 points. Jan De Beule, Ghent University

Optimal codes and related topics 2005 6 More parameters We consider now { ǫ 2 ( q 2 + q + 1) + ǫ 1 ( q +1)+ ǫ 0 , ǫ 2 ( q +1)+ ǫ 1 ; k, q } -minihypers ( F, w ) , ǫ 2 + ǫ 1 + ǫ 0 � √ q , k � 3 . Using the results on { ǫ 1 ( q + 1) + ǫ 0 , ǫ 1 ; k, q } -minihypers, and using an induction hypothesis, we prove: An { ǫ 2 ( q 2 + q + 1) + ǫ 1 ( q + Theorem 4. 1) + ǫ 0 , ǫ 2 ( q + 1) + ǫ 1 ; k − 1 , q } -minihyper ( F, w ) , ǫ 2 + ǫ 1 + ǫ 0 � √ q , k � 4 , is a sum of ǫ 2 planes, ǫ 1 lines and ǫ 0 points. Jan De Beule, Ghent University

Optimal codes and related topics 2005 7 The general case We consider a t t � � { ǫ i v i ; k, q }− minihyper( F, w ) ǫ i v i +1 , i =0 i =1 i =0 ǫ i � √ q k � 2 , 1 � t < k, � t Using an induction hypothesis on k , and the obtained characterisation results for smaller t , we can prove that ( F, w ) is a sum of ǫ t t -dimensional subspaces PG( t, q ) , ǫ t − 1 t − 1 -dimensional subspaces PG( t − 1 , q ) , . . . , ǫ 1 lines and ǫ 0 points. Jan De Beule, Ghent University

Optimal codes and related topics 2005 8 References [1] A. Blokhuis, L. Storme, and T. Sz˝ onyi. Lacunary polynomials, multiple blocking sets and Baer subplanes. J. London Math. Soc. (2) , 60(2):321–332, 1999. [2] A. Blokhuis, L. Storme, and T. Sz˝ onyi. Lacunary polynomials, multiple blocking sets and Baer subplanes. J. London Math. Soc. (2) , 60(2):321–332, 1999. Jan De Beule, Ghent University