Questions tagged [finite-geometry]
Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.
65 questions
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An upper bound on the number of sets of parallel lines covering points in a finite plane?
Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...
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$\left< 15\right>^7/15$-womcode construction
In the article Womcodes constructed with projective geometries Frans Merkx constructed several good wom-codes (write-once memory codes, see How to reuse a "write-once" memory by Rivest & Shamir ...
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Applications of finite Bolyai-Lobachevsky planes
Google scholar gives more than 200 articles comcerning finite Bolyai-Lobachevsky (BL) planes. Usually they devoted to construction of such objects (axioms may be different).
Are their any ...
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Intersection of two trace equations over finite fields
Let $F_q$ be a finite field with $q$ elements. Let $n$ be an integer and $Tr:F_{q^n} \rightarrow F_q$ the trace function. My question is: For which integer $k$,
$$\{x: Tr(x)=0\}\cap\{x: Tr(x^k)=0\}=\{...
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Vector spaces over a field of prime order with certain hyperplanes
Let $V$ be a vector space of finite dimentional $d$ over a field of prime order $p$.
For what values of $d$ and $p$, one can find $d+1$ (pairwise distinct) hyperplanes (subspaces of dimension $d-1$) $...
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For which finite projective planes can the incidence structure be written as a circulant matrix?
It is well known that the projective plane of order $2$ can be represented by the circulant matrix $M_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix}
x&x&1&x&1&1&1\\
1&x&x&...
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Is inner product preserved only by the stabiliser in a finite reflection group?
Is the following statement true for finite reflection groups?
Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$
and let $z$ be in the orbit of $y$. If $\...
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Different powers of a primitive root simultaneously lying in a subspace
Let $p$ be a large prime and let $\alpha$ be a root of a primitive quadratic polynomial over $\mathbb{F}_p$. Let $N$ be an integer parameter of size proportional to $p$ and $$V = \{\alpha + b : b \in \...
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Is the finite projective plane stable as an extremal set system?
Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
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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
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Enumerating certain types of permutation polynomials
Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all ...
- 1,197
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A problem in Galois Geometry
Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $w^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2w^{k}$ given ...
- 13.9k
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Finding a set of disjoint affine subspaces such that their union is equal to a given subset of $\mathbb{F}_2^n$
Suppose I'm given a set of point $S = \{x_1, \dots, x_m \} \subseteq \mathbb{F}_2^n$, and the following task. Find a set of disjoint affine subspaces of $\mathbb{F}_2^n$, $A_1, \dots, A_k$ satisfying ...
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Dual of blocking sets in finite geometry
Let $V$ be an $n$-dimensional vector space over the finite field of cardinality $q$ and let $W_1,\ldots,W_m$ be hyperplanes of $V$ such that
$$V=\bigcup_{i=1}^mW_i \,\,\hbox{ and }\,\,0=\bigcap_{i=...
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Elliptic Curve, characteristic equation of Frobenius endomorphism relation to isogeny
Let E be an elliptic curve over $F_p$. Suppose that its j invarient is not supersingular and that $j\neq 0 $ or 1728.
Then the modular polynomial $\Phi_l(j,T)$ has a zero $\tilde{\jmath} \in \...
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