164 problems

Conjecture on the 2-rank of $M_2$.

For every positive integer $n$, every positive integer $k$ with $\gcd(k,n)=1$, and every finite field $K$ of characteristic $2$ with $|K|=2^n$, let $d_k=4^k-2^k+1$ and…

For every linear code $\mathcal{C}\subseteq\mathbb{F}_q^n$ of dimension $k$ over every finite field $\mathbb{F}_q$, and every admissible order $t$ with $1\le t\le k$, the $t$-th ge…

For each integer $n\ge 0$, determine whether there exists a bijection $A:\{0,1\}^n\to\{1,\ldots,2^n\}$ such that, for every pair of distinct coordinates $i,j\in\{1,\ldots,n\}$ and…

Let $\mathcal{U}$ be a linear code in the projective space $\mathbb{P}_q(n)$. An indecomposable basis is a basis of the vector space over $\mathbb{F}_2$ formed by the code whose el…

Let $q$ be a prime power, let $\mathcal{Q}=Q^+(5,q)$ be the Klein quadric, and fix a plane $\pi$ in $\mathcal{Q}$. Let $\mathcal{P}$ be the set of planes in $\mathcal{Q}$ disjoint…

Let $V$ be an $n$-dimensional vector space over $\mathbb{F}_q$, and let $k \ge d \ge 3$ be integers satisfying … For a family $\mathcal{F} \subset \genfrac{[}{]}{0pt}{}{V}{k}$, cal…

Let $q$ be a prime power, and let the collinearity graph of the polar space $\mathcal{Q}^{-}(5,q)$ be the graph whose vertices are the points of $\mathcal{Q}^{-}(5,q)$, with two ve…

Let $q$ be a power of $3$, let $G=PGL_2(q)$ act on the lines of $PG(3,q)$, and let $C$ be the twisted cubic. A line is generic if it neither intersects $C$ nor lies in any osculati…

Let $q$ be a prime power, let $F\in\mathbb{F}_{q^2}[x_0,x_1,x_2,x_3]$ be homogeneous of degree $d$, and let $V_2\subset\mathbb{P}^3$ be a non-degenerate Hermitian surface defined o…

Let $m\ge 8$ be an even integer, let $b,c\in\mathbb{F}_{2^m}$, and let $r$ satisfy $r\le m/2$ and $\gcd(r,m)=1$. Let $f_{b,c,r}$ denote the function defined in Theorem. Non-APN con…

Let $M$ be a matroid, and call a rank-$k$ flat ordinary if it is the direct sum of a point and a rank-$(k-1)$ flat. Ordinary-flat conjecture. For each integer $k\ge 2$, every compl…

Let $V=\mathbb{F}_q^n$ and let $J_q(n,k)$ be the Grassmann graph whose vertices are the $k$-dimensional subspaces of $V$. A function $f$ on the vertices is Boolean degree $1$ if it…

Let $A_q(n,d;k)$ denote the maximum size of a constant-dimension $k$-subspace code in $\mathbb{F}_q^n$ with minimum subspace distance at least $d$. Generalized vector space partiti…

Bonin's conjecture. The matroid $M$ has at most

Let ${\mathcal D}$ be a symmetric design with Singer parameters … where $q=p^s$ and $p$ is prime. Write ${\rm rank}_p{\mathcal D}$ for the dimension over ${\mathbb F}_p$ of the cod…

Let $\sigma:{\mathbb{F}_q}\rightarrow{\mathbb{F}_q}$ be a permutation, and let $\Gamma_\sigma=\{(x,\sigma(x)):x\in{\mathbb{F}_q}\}\subset{\mathbb{F}_q}^2$ be its graph. A collinear…

For an odd prime power $q$, a Besicovitch set in ${\mathbb{F}_q}^2$ is a subset containing a line in every direction. Finite-plane Besicovitch set conjecture. The smallest Besicovi…

Maximum subgraph conjecture. If $U$ spans a complete subgraph or an empty subgraph of order $q$ in $V_q$, then $U$ is a line in $F_q\times F_q$.

Hollmann's conjecture. If $m$ is an odd prime, then $({\cal E},\{\Delta_a\})$ is pseudocyclic.

Codimension conjecture. The codimension of

Let $n>2$ be prime, and define a permutation $g$ of $\mathbb Z_n$ by … Order permutations lexicographically, and let $\Psi(n)$ be the minimum number of collinear triples in a permu…

Let $q$ be an odd prime power, let $\beta$ be a primitive element of $\mathbb F_{q^2}$, and define … for $r\in\mathbb F_q$. Set … and let $\mathcal U_{\beta}$ be the…

Two-barrier Ore-degree conjecture. If

Prime affine direction conjecture.