All Questions
Tagged with finite-geometry linear-algebra
12 questions
7
votes
0
answers
301
views
Arrangement of subspaces over finite fields
I'm trying to find out what is already known about the following setup.
Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
- 1,062
6
votes
1
answer
297
views
Covering the finite plane with lines
This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...
- 23k
4
votes
0
answers
192
views
Forcing scalar products to avoid prescribed values
Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition:
\begin{gather}
\text{For ...
- 23k
2
votes
1
answer
302
views
For which finite projective planes can the incidence structure be written as a circulant matrix?
It is well known that the projective plane of order $2$ can be represented by the circulant matrix $M_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix}
x&x&1&x&1&1&1\\
1&x&x&...
- 13.4k
3
votes
1
answer
164
views
A system of linear equations related to the geometry of a finite plane
Let $\mathcal L$ denote the set of all lines in $\mathbb F_p^2$ parallel to one of the lines
$$ X:=\{(x,0)\colon x\in\mathbb F_p \},
\ Y:=\{(0,y)\colon y\in\mathbb F_p \},
\ Z:=\{(z,z)\...
- 23k
6
votes
1
answer
303
views
Maximum number of elements in union of subspaces
Let $V$ be a $m$-dimensional vector space over $\mathbb{F}_q$ and $1<\ell<m-1$. Let $r$ be a positive integer such that $r\ell\leq m$.
QUESTION. What is the maximum number of elements in the ...
- 71
3
votes
3
answers
611
views
On MDS code property
Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...
- 13.9k
2
votes
0
answers
86
views
A system of homogeneous linear equations
This is the "real-life" (but slightly more technical) version of a question I have asked recently.
For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of ...
- 23k
4
votes
2
answers
383
views
Finding the set of all $0$-$1$ vectors in an affine subspace
We are given a $0$-$1$ matrix $A$ with constant row and column sum, and we need to find out if there exists a $0$-$1$ vector in the solution space of $Ax = \mathbf{1}$ over $\mathbb{Q}$ (or $\mathbb{Z}...
- 1,197
10
votes
2
answers
495
views
Is there a hyperplane avoiding two independent sets?
Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
- 11.3k
5
votes
0
answers
235
views
A question on hyperplanes in partial linear spaces and hypergraphs
A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear ...
- 1,197
3
votes
0
answers
195
views
Vector spaces over a field of prime order with certain hyperplanes
Let $V$ be a vector space of finite dimentional $d$ over a field of prime order $p$.
For what values of $d$ and $p$, one can find $d+1$ (pairwise distinct) hyperplanes (subspaces of dimension $d-1$) $...
- 3,738