The complete characterization of the minimum size supertail - PowerPoint PPT Presentation

Introduction The Supertail Main Theorem Future Work The complete characterization of the minimum size supertail Esmeralda L. N˘ astase Xavier University Joint work with P. Sissokho Illinois State University May 18, 2019 E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size ◮ V = V ( n , q ) the vector space of dimension n over GF ( q ). ◮ A subspace partition or partition P of V , is a collection of subspaces { W 1 , . . . , W k } s.t. ◮ V = W 1 ∪ · · · ∪ W k ◮ W i ∩ W j = { 0 } for i � = j . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size ◮ V = V ( n , q ) the vector space of dimension n over GF ( q ). ◮ A subspace partition or partition P of V , is a collection of subspaces { W 1 , . . . , W k } s.t. ◮ V = W 1 ∪ · · · ∪ W k ◮ W i ∩ W j = { 0 } for i � = j . ◮ size of a subspace partition P is the number of subspaces in P . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Applications ◮ translation planes ◮ error-correcting codes ◮ orthogonal arrays ◮ designs ◮ subspace codes E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Let P be any partition of V . m d 1 m dk ◮ P has type d . . . d , if for each i , there are m d i > 0 1 k subspaces of dim d i in P , and d 1 < d 2 < · · · < d k . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Let P be any partition of V . m d 1 m dk ◮ P has type d . . . d , if for each i , there are m d i > 0 1 k subspaces of dim d i in P , and d 1 < d 2 < · · · < d k . Problem ◮ What are the necessary and sufficient conditions for the existence of a partition of V of a given type? E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Every partition P of V satisfies: ◮ packing condition k m d i ( q d i − 1) = q n − 1 � i =1 E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Every partition P of V satisfies: ◮ packing condition k m d i ( q d i − 1) = q n − 1 � i =1 ◮ dimension condition U , W ∈ P , U � = W = ⇒ dim( U ) + dim( W ) ≤ n E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Let P be a partition of V . ◮ σ q ( n , s ) = the min size of any partition of V in which the largest subspace has dim s . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Definitions The Supertail Applications Main Theorem Motivation Future Work The Minimum Size Theorem (Heden, Lehmann, N., and Sissokho, 2011, 2012). Let n , m , s , and r be integers such that 1 ≤ r < s , m ≥ 1, and n = ms + r . Then q s + 1 for 3 ≤ n < 2 s ,      σ q ( n , s ) = m − 2 q is + q ⌈ s + r 2 ⌉ + 1 for n ≥ 2 s .  q s + r �    i =0 E. N˘ astase The complete characterization of the minimum size supertail

Introduction The Supertail Results Main Theorem The Minimum Size Future Work m dk m d 1 Let P be a partition of V of type d . . . d . 1 k ◮ For any s such that d 1 < s ≤ d m , the set S of subspaces in P of dim less than s and with greatest subspace dim t is called the st -supertail of P . E. N˘ astase The complete characterization of the minimum size supertail

Introduction The Supertail Results Main Theorem The Minimum Size Future Work Theorem (Heden, Lehmann, N., and Sissokho, 2013). Let P be a partition of V . If S is an st -supertail of P , then |S| ≥ σ q ( s , t ) . E. N˘ astase The complete characterization of the minimum size supertail

Introduction The Supertail Results Main Theorem The Minimum Size Future Work Theorem (Heden, Lehmann, N., and Sissokho, 2013). Let P be a partition of V . If S is an st -supertail of P , then |S| ≥ σ q ( s , t ) . Corollary. If s ≥ 2 t and |S| = σ q ( s , t ), then the union of the subspaces in S forms a subspace of dim s . E. N˘ astase The complete characterization of the minimum size supertail

Introduction The Supertail Results Main Theorem The Minimum Size Future Work Conjecture (2013). If t < s < 2 t and |S| = σ q ( s , t ) = q t + 1, then the union of the subspaces in S forms a subspace. E. N˘ astase The complete characterization of the minimum size supertail

Introduction The Supertail Results Main Theorem The Minimum Size Future Work Conjecture (2013). If t < s < 2 t and |S| = σ q ( s , t ) = q t + 1, then the union of the subspaces in S forms a subspace. Theorem (Heden, 2009). Let P be a partition of V of type m d 1 m dk d . . . d . If S is the tail of P , i.e., all subspaces in S have the 1 k same dim d 1 = t , s.t. |S| = q t + 1 and d 2 = s < 2 t , then the subspaces of S form a subspace of dim 2 t . E. N˘ astase The complete characterization of the minimum size supertail

Introduction The Supertail Results Main Theorem The Minimum Size Future Work Theorem (N. and Sissokho, 2017). Let P be a partition of V of m d 1 m dk type d . . . d , and let S be an st -supertail of P s.t. 1 k |S| = σ q ( s , t ) and t < s < 2 t . If one of the following conditions holds (i) S contains subspaces of at most 2 different dimensions (ii) s = 2 t − 1 (iii) All the subspaces in P \ S have the same dimension s , then the union of the subspaces in S forms a subspace W , and either (a) d 1 = t , m d 1 = q t + 1, and dim W = 2 t , or (b) d 1 = a and d 2 = t , with m d 1 = q t and m d 2 = 1, and dim W = a + t . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Notation The Supertail Auxiliary Results Main Theorem Proof Future Work Final Result Theorem (N. and Sissokho, 2018). Let P be a partition of V . Suppose S is an st -supertail of P such that t < s < 2 t and |S| = σ q ( s , t ) = q t + 1 . Then (i) the union of the subspaces in S forms a subspace W (ii) S is a subspace partition of W whose type is either ◮ t q t +1 , or ◮ t 1 a q t , for some 1 ≤ a < t . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Notation The Supertail Auxiliary Results Main Theorem Proof Future Work Final Result Let P be a partition of V . 1. Let Θ k = ( q k − 1) / ( q − 1) ⇒ Θ k is the the number of points, i.e., 1-subspaces, in an k -subspace. E. N˘ astase The complete characterization of the minimum size supertail

Introduction Notation The Supertail Auxiliary Results Main Theorem Proof Future Work Final Result Let P be a partition of V . 1. Let Θ k = ( q k − 1) / ( q − 1) ⇒ Θ k is the the number of points, i.e., 1-subspaces, in an k -subspace. 2. Let H denote the set of all hyperplanes of V . For H ∈ H , and any integer k ≥ 1, ◮ b H , k =the number of k -subspaces X ∈ S such that X ⊆ H . ◮ β H = � t k = a b H , k q k , where a ≤ dim X ≤ t for any X ∈ S . E. N˘ astase The complete characterization of the minimum size supertail

Introduction Notation The Supertail Auxiliary Results Main Theorem Proof Future Work Final Result Lemma 1. The number of hyperplanes H ∈ H that contain a given k -subspace of V is Θ n − k = q n − k − 1 . q − 1 In particular, H contains Θ n = q n − 1 q − 1 hyperplanes. E. N˘ astase The complete characterization of the minimum size supertail

Introduction Notation The Supertail Auxiliary Results Main Theorem Proof Future Work Final Result Lemma 2. Let P be a partition of V . Suppose S is an st -supertail of P such that |S| = σ q ( s , t ) = q t + 1 , and t < s < 2 t . If H ∈ H , then t m i Θ i = cq s − 1 β H ≥ q t and � q − 1 i = a for some integer c ≥ 1. E. N˘ astase The complete characterization of the minimum size supertail

Introduction Notation The Supertail Auxiliary Results Main Theorem Proof Future Work Final Result Lemma 3. Let P be a partition of V . Suppose S is an st -supertail of P such that |S| = σ q ( s , t ) = q t + 1 and t < s < 2 t . Let W = � X ∈S X and let δ = δ ( S ) denote the number of points, i.e., 1-subspaces, of W . For H ∈ H , let δ H = δ H ( S ) be the number of points in W ∩ H . Then (i) |S| − 1 = q t ≤ β H ≤ cq d + q t = δ ( q − 1) + |S| (ii) � H δ H = δ Θ n − 1 (iii) � H δ H ( δ H − 1) = δ ( δ − 1)Θ n − 2 . (iv) β H = q δ H − δ + |S| . (v) � H β H = |S| Θ n − δ . |S| 2 + δ ( q − 1) H β 2 − δ 2 ( q − 1) − δ (2 |S| − 1). (vi) � � � H = Θ n (vii) � H ( β H − ( |S| − 1)) ( β H − ( δ ( q − 1) + |S| )) = 0. E. N˘ astase The complete characterization of the minimum size supertail