# Tagged Questions

Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.

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What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line? Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,... 1answer 245 views ###$(n-2)$-blocking sets in$AG(n,2)$Let's define$k$-blocking set in affine space$AG(n,q)$a set that meets every coset (translate of subspace) of dimension$k$. I have seen a lot work related to minimal$(n-1)$-blockings set. ... 0answers 66 views ### Is there a general theory for the structure of the (semi)group generated by morphisms of an affine space$F_p^3$? Consider an affine space$\mathbb{F}_p^3$, and assume we have a handful of morphisms$f_i : \mathbb{F}_p^3 \rightarrow \mathbb{F}_p^3$given by $$f_i(x, y, z) =(P_i(x, y, z), Q_i(x, y, z), R_i(x, y, z)... 1answer 161 views ### 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 ... 0answers 106 views ### The Maximum Number of Lines Contained in the Point Set of a Finite Projective Plane Consider a finite projective plane of order q. Define f(m) to be the maximum number of lines completely contained in any point set of size m, where 1 \leq m \leq q^2+q+1. I would like to ... 0answers 137 views ### Applications of small Kakeya sets over finite fields It was proved by Dvir that a Kakeya set in \mathbb{F}_q^n has size at least q^n/n!, a bound which was later improved to q^n/2^n. For n = 2 and q odd the exact bound is q(q+1)/2 + (q-1)/2 ... 0answers 177 views ### Sets of spreads in graphs Let G be a graph. A k-spread is a set of cliques of order k which partition the vertex set (so k|n, where n is the number of vertices). A partial k-resolution of G is a set of pairwise ... 2answers 846 views ### How close can one get to the missing finite projective planes? This question can be interpreted as an instance of the Zarankiewicz problem. Suppose we have an n\times n matrix with entries in \{0,1\} with no \begin{pmatrix}1 & 1\\ 1& 1\end{pmatrix} ... 1answer 154 views ### 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 \... 2answers 389 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? 0answers 176 views ### Subplanes of Finite Projective Planes If a finite projective plane \pi_1 of order m contains, as a sub plane, a finite projective plane \pi_2 of order n, then m \geq n^2 with equality holding only in the case of a Baer sub plane.... 2answers 331 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}... 0answers 295 views ### 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 ... 0answers 141 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 ... 1answer 150 views ### Point-Hyperplane incidence in finite projective spaces Let P be a finite projective space of order q and dimension d. I am interested in finding the least k such that for any set S of k points of P, and for any set S' of k hyperplanes of ... 0answers 92 views ### 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\}=\{... 2answers 296 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 ... 0answers 234 views ### 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 ... 2answers 554 views ### Who constructed the projective plane of order$4$from$K_6$? I have been trying to hunt down the original reference for the construction of the projective plane of order$4$from the complete graph on$6$vertices. The reference I have at hand are Cameron and ... 2answers 160 views ### Incidence matrices of generalized quadrangles Is there somewhere a database of incidence matrices of generalized quadrangles that one can download? 1answer 231 views ### Covering all, but$k$points with affine subspaces For non-negative integer$d\le n$and$k\le 2^n$, how many affine subspaces of co-dimension$d$are needed to cover all, but exactly$k$elements of the vector space${\mathbb F}_2^n$, and what are ... 2answers 664 views ### On the joints problem in finite fields The original version of the so-called "joints problem" consists of the following: Let$L$be a set of lines in$\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, ... 0answers 179 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$d-1$)$...
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved. ...