scholarly journals Constant Rank-Distance Sets of Hermitian Matrices and Partial Spreads in Hermitian Polar Spaces

10.37236/3534 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Rod Gow ◽  
Michel Lavrauw ◽  
John Sheekey ◽  
Frédéric Vanhove
Keyword(s):  
Maximum Size ◽  
General Notion ◽  
Partial Spread ◽  
Constant Rank ◽  
Hermitian Matrices ◽  
Rank Distance ◽  
Partial Spreads ◽  
Related Notion ◽  
Distance Set ◽  
Distance Sets

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes.

scholarly journals The maximum size of a partial spread in a finite projective space

2017 ◽  
Vol 152 ◽  
pp. 353-362 ◽  
Author(s):  
Esmeralda L. Năstase ◽  
Papa A. Sissokho
Keyword(s):  
Projective Space ◽  
Maximum Size ◽  
Finite Projective Space ◽  
Partial Spread

scholarly journals Constructions of Maximum Few-Distance Sets in Euclidean Spaces

10.37236/8565 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ferenc Szöllősi ◽  
Patric R.J. Östergård
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Dimensional Euclidean Space ◽  
Combined Approach ◽  
Euclidean Spaces ◽  
Grobner Basis ◽  
Finite Set ◽  
Distance Set ◽  
Distance Sets ◽  
Basis Computation

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

Maximal Strictly Partial Spreads

1978 ◽  
Vol 30 (03) ◽  
pp. 483-489 ◽  
Author(s):  
Gary L. Ebert
Keyword(s):  
Projective Space ◽  
Partial Spread ◽  
3 Dimensional ◽  
Partial Spreads ◽  
Maximal Partial Spread ◽  
Dimensional Projective Space

Let ∑ = PG(3, q) denote 3-dimensional projective space over GF(q). A partial spread of ∑ is a collection W of pairwise skew lines in ∑. W is said to be maximal if it is not properly contained in any other partial spread. If every point of ∑ is contained in some line of W, then W is called a spread. Since every spread of PG(3, q) consists of q2 + 1 lines, the deficiency of a partial spread W is defined to be the number d = q2 + 1 — |W|. A maximal partial spread of ∑ which is not a spread is called a maximal strictly partial spread (msp spread) of ∑.

Distance Sets of Urysohn Metric Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 222-240 ◽  
Author(s):  
N.W. Sauer
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Finite Metric Space ◽  
Distance Set ◽  
Distance Sets ◽  
Separable Metric Space

Abstract.A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

scholarly journals Bounds on Antipodal Spherical Designs with Few Angles

10.37236/9891 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Zhiqiang Xu ◽  
Zili Xu ◽  
Wei-Hsuan Yu
Keyword(s):  
Unit Sphere ◽  
Finite Subset ◽  
Upper Bound ◽  
Maximum Size ◽  
Tight Frames ◽  
Previous Estimate ◽  
Spherical Designs ◽  
Equiangular Tight Frames ◽  
Special Cases ◽  
Distance Set

A finite subset $X$ on the unit sphere $\mathbb{S}^d$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X, \mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set $X$ for small $s$. Motivated by the method developed by Nozaki and Suda, for each even integer $s\in[\frac{t+5}{2}, t+1]$ with $t\geq 3$, we improve the best known upper bound of Delsarte, Goethals and Seidel. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

scholarly journals The maximum size of a partial spread in H(5,q2) is q3+1

2007 ◽  
Vol 114 (4) ◽  
pp. 761-768 ◽  
Author(s):  
J. De Beule ◽  
K. Metsch
Keyword(s):  
Maximum Size ◽  
Partial Spread

scholarly journals The Maximum Size of a Partial Spread in $H(4 n +1, q^2)$ is $q^{2 n +1}+1$

10.37236/251 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Frédéric Vanhove
Keyword(s):  
Maximum Size ◽  
Polar Space ◽  
Maximal Dimension ◽  
Partial Spread ◽  
Isotropic Subspaces ◽  
Hermitian Polar Space ◽  
Totally Isotropic Subspaces

We prove that in every finite Hermitian polar space of odd dimension and even maximal dimension $\rho$ of the totally isotropic subspaces, a partial spread has size at most $q^{\rho+1}+1$, where $GF(q^2)$ is the defining field. This bound is tight and is a generalisation of the result of De Beule and Metsch for the case $\rho=2$.

scholarly journals POINTS ON HYPERBOLAS AT RATIONAL DISTANCE

2012 ◽  
Vol 08 (04) ◽  
pp. 911-922
Author(s):  
EDRAY HERBER GOINS ◽  
KEVIN MUGO
Keyword(s):  
Elliptic Curve ◽  
Rational Numbers ◽  
Rational Points ◽  
Conic Section ◽  
Distance Set ◽  
Set Of Points ◽  
Distance Sets

Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article, we consider such a set of rational points restricted to a given hyperbola. To be precise, for rational numbers a, b, c, and d such that the quantity D = (ad - bc)/(2a2) is defined and non-zero, we consider rational distance sets on the conic section axy + bx + cy + d = 0. We show that, if the elliptic curve Y2 = X3 - D2X has infinitely many rational points, then there are infinitely many sets consisting of four rational points on the hyperbola such that their pairwise distances are rational numbers. We also show that any rational distance set of three such points can always be extended to a rational distance set of four such points.

Distance Sets

1951 ◽  
Vol 3 ◽  
pp. 187-194 ◽  
Author(s):  
L. M. Kelly
Keyword(s):  
Distance Set ◽  
Set Of Points ◽  
Distance Sets

With each set of points S of a distance space there is associated a set of non-negative real munbers D(S) called the distance set of 5. The number x is an element of D(S) if and only if x is a distance between some pair of points of 5. The number zero is always an element of any distance set and no two distinct elements are equal.

Spreads which are Not Dual Spreads

1969 ◽  
Vol 12 (6) ◽  
pp. 801-803 ◽  
Author(s):  
A. Bruen ◽  
J. C. Fisher
Keyword(s):  
Projective Space ◽  
Pairwise Disjoint ◽  
Partial Spread ◽  
Partial Spreads

In this note we show the existence of a spread which is not a dual spread, thus answering a question in [1]. We also obtain some related results on spreads and partial spreads.Let ∑ = PG(2t-l, F) be a projective space of odd dimension (2t-l, ≥2) over the field F. In accordance with [1] we make the following definitions. A partial spread S of ∑ is a collection of (t-l)-dimensional projective subspaces of ∑ which are pairwise disjoint (skew).

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