Intersection patterns of linear subspaces with the hypercube

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:164

Article

Melo, Nolmar¹  Winter, Andreas^2,3 

[1] Univ Fed Vales Jequitinhonha & Mucuri, Dept Ciencias Exatas, BR-39803371 Teofilo Otoni, MG, Brazil

[2] ICREA, Pg Lluis Co 23, ES-08001 Barcelona, Spain

[3] Univ Autonoma Barcelona, Grup Informat Quant, Dept Fis, ES-08193 Bellaterra, Barcelona, Spain

关键词: Algebraic combinatorics;    Hypercube;    Linear subspace;   

DOI  :  10.1016/j.jcta.2018.12.006

来源: Elsevier

【 摘 要 】

Following a combinatorial observation made by one of us recently in relation to a problem in quantum information Nakata et al. (2017) [7], we study what are the possible intersection cardinalities of a k-dimensional subspace with the hypercube in n-dimensional Euclidean space. We also propose two natural variants of the problem by restricting the type of subspace allowed. We find that whereas every natural number eventually occurs as the intersection cardinality for some k and n, on the other hand for each fixed k, the possible intersections sizes are governed by severe restrictions. To wit, while the largest intersection size is evidently 2(k), there is always a large gap to the second largest intersection size, which we find to be 3/4 2(k) for k >= 2 (and 2(k-1) in the restricted version). We also present several constructions, and propose a number of open questions and conjectures for future investigation. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free