Higgledy-piggledy subspaces and uniform subspace designs

Fancsali, Szabolcs Levente and Sziklai, Péter (2016) Higgledy-piggledy subspaces and uniform subspace designs. DESIGNS CODES AND CRYPTOGRAPHY, 79 (3). pp. 625-645. ISSN 0925-1022

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Abstract

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.

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