8 problems

Ward's divisibility conjecture. Suppose that $p^e\mid d$ and $p^e\geq q$. Then $p^{e+1}/q$ is a divisor of $C$.

Binary minimum-length conjecture. For $r\geq 2$,

Divisible-set union conjecture. There exists a function $f\colon \mathbb{N}\to\mathbb{N}$ such that every $2^r$-divisible set of $f(r)\cdot [r+1]_2$ points over $\mathbb{F}_2$ is t…

Let $q$ be a prime power, let $r$ and $v$ be integers, and let $\mathcal{S}$ be a $q^r$-divisible set of $q^{r+1}$ points in $\operatorname{PG}(v-1,q)$. An $(r+1)$-cylinder is the…

Let $q$ be a prime power, let $r$ and $v$ be non-negative integers, and consider $q^r$-divisible spanning sets of $q^{r+1}$ points in $\operatorname{PG}(v-1,q)$. The generalized cy…

Let $v$ and $r$ be non-negative integers and let $q$ be a prime power. A set $\mathcal{S}$ of points in $\operatorname{PG}(v-1,q)$ is called $q^r$-divisible if every hyperplane mee…

Let $\mathcal{S}$ be a set of $p^2$ points in $\operatorname{AG}(3,p)$, where $p$ is prime. Suppose that every plane is incident with a number of points of $\mathcal{S}$ congruent…

Let $C$ be a projective $9$-divisible linear code over $\boldsymbol{F}_3$ with parameters $[70,7]_3$, meaning that every codeword weight is divisible by $9$ and no two columns of a…