Properties of sets of subspaces with constant intersection dimension

Lisa Hernandez Lucas

doi:10.3934/amc.2020052

Article Contents

2021, Volume 15, Issue 1: 191-206. Doi: 10.3934/amc.2020052

This issue Previous Article Repeated-root constacyclic codes of length

$ 6lp^s $

Lisa Hernandez Lucas^,

Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

^* Corresponding author: Lisa Hernandez Lucas
^* Corresponding author: Lisa Hernandez Lucas

Received: April 2019

Revised: October 2019

Early access: January 2020

Published: February 2021

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Abstract

A $ (k,k-t) $-SCID (set of Subspaces with Constant Intersection Dimension) is a set of $ k $-dimensional vector spaces that have pairwise intersections of dimension $ k-t $. Let $ \mathcal{C} = \{\pi_1,\ldots,\pi_n\} $ be a $ (k,k-t) $-SCID. Define $ S: = \langle \pi_1, \ldots, \pi_n \rangle $ and $ I: = \langle \pi_i \cap \pi_j \mid 1 \leq i < j \leq n \rangle $. We establish several upper bounds for $ \dim S + \dim I $ in different situations. We give a spectrum result under certain conditions for $ n $, giving examples of $ (k,k-t) $-SCIDs reaching a large interval of values for $ \dim S + \dim I $.

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Mathematics Subject Classification: Primary: 51E20; Secondary: 05B25, 51E23, 94B60.

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Table 1. Summary of the best bounds found for $\dim S +\dim I$, for different values of $n$, $k$ and $t$

Condition	Upper bound $\dim S + \dim I$	Sharp?	Theorem
$(k-t)(n-1) \leq k$	$nk$	yes	Theorem 2.1 & 2.2
$k\geq 2t$, $n\geq 3$,$(k,n)\neq(2t,3)$	$2k+2(n-2)t-(n-3)$	unknown	Theorem 2.5
$k <2t$ $(k-t)(n-1) > k$	$nk$	no	Theorem 2.1 & 2.2

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