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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 …

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 $q(q+1)/2 + (q-1)/2$ …

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 …

Not much is known for the general case. Let $m(k, n, q)$ denote the minimum size of an $k$-blocking set in $AG(n, q)$. Trivially we have $m(0, n, q) = q^n$ and $m(n, n, q) = 1$. By Jamison/Brouwer-S …

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 …

Here is an improvement of the upper bound which I found in ``The polynomial method in Galois geometries'' by Simeon Ball. See page number 4. The known constructions are somewhat crude. For exampl …

It has been proved by Simeon Ball that for $k \leq p$, all $[n, k, n-k+1]_q$ codes are Reed-Solomon codes, where $q = p^h$. See Corollary 9.2 in the following paper: Ball, S. On sets of vectors of a …

Here are some partial answers to your question. Let $A = A_1 \times \dots \times A_n \subseteq F^n$ be a finite grid. Alon and Furedi proved that you need at least $\sum (\# A - 1)$ hyperplanes to …

This recent paper of Sato and Suzuki shows that the graphs corresponding to some classical generalized quadrangles are indeed Hamiltonian: Sato, H. & Suzuki, H. Graphs and Combinatorics (2018). http …

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 al …