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Higgledy-piggledy subspaces and uniform subspace designs

机译：杂乱无章的子空间和统一子空间设计

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In this article, we investigate collections of ` well-spread-out' projective (and linear) subspaces. Projective k-subspaces in PG(d, F) are in ` higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension k in a generator set of points. We prove that the higgledy-piggledy set H of k-subspaces has to contain more than min {vertical bar F vertical bar, Sigma(k)(i=0) left perpendicular d-k+I/i+1 right perpendicular} elements. We also prove that H has to contain more than (k + 1) . (d -k) elements if the field F is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces H-1 ,..., H-N <= F-m each of rank r such that each linear subspace W <= F-m of rank s meets at most A among them. This subspace design is an r-uniform strong (s, A) subspace design if Sigma(N)(i=1) rank(H-i boolean AND W) <= A for for all W <= F-m of rank s. We prove that if m = r + s then the dual ({H-1(perpendicular to) ,..., H-N(perpendicular to)}) of an r-uniform weak (strong) subspace design of parameter (s, A) is an s-uniform weak (strong) subspace design of parameter (r, A). We show the connection between uniform weak subspace designs and higgledy-piggledy subspaces proving that A >= min { vertical bar F vertical bar, Sigma(r=1)(i=0) left perpendicular s+i/i+1right perpendicular} for r-uniform weak or strong (s, A) subspace designs in Fr+ s. We show that the r-uniform strong (s, r . s + ((r)(2))) subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter A = r .s if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound (k + 1) . (d -k) + 1 over algebraically closed field is tight.
机译：在本文中，我们研究了“分布良好”的射影（和线性）子空间的集合。如果PG（d，F）中的投影k-子空间满足点生成器集合中共维数k的每个投影子空间，则它们处于“杂草丛生”的状态。我们证明k个子空间的hi杂散集H必须包含多个min {垂直条F垂直条，Sigma（k）（i = 0）左垂直d-k + I / i + 1右垂直}元素。我们还证明H必须包含大于（k +1）的元素。 （d -k）元素，如果字段F是代数封闭的。 r均匀弱（s，A）子空间设计是每个等级r的线性子空间H-1，...，HN <= Fm的集合，使得等级s的每个线性子空间W <= Fm最多满足A其中。对于所有s≤W-F-m的Sigma（N）（i = 1）rank（H-i布尔AND W）<= A的情况，此子空间设计是r均匀强（s，A）子空间设计。我们证明，如果m = r + s，则参数（s，A）的r均匀弱（强）子空间设计的对偶（{H-1（垂直于，，...，HN（垂直于）}}） ）是参数（r，A）的s均匀弱（强）子空间设计。我们证明了均匀的弱子空间设计与杂乱无章的子空间之间的联系，证明了A> = min {垂直条F垂直条，Sigma（r = 1）（i = 0）垂直s + i / i + 1垂直} Fr + s中的r均匀弱或强（s，A）子空间设计。我们证明了由Guruswami和Kopparty（基于多重性代码）构造的r均匀强（s，r .s +（（r）（2）））子空间设计具有参数A = r .s（如果我们将其视为a）弱子空间设计。我们给出了弱子空间设计和强子空间设计（以及杂乱无章的子空间）的一些类似构造，并证明了下界（k +1）。 （d -k）+ 1在代数闭合域上是紧的。

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