22 problems

Let ${\textup{PG}}(2,q^3)$ be a projective plane with $q$ even and $q>2$. Let $\mathcal{C}$ be a conic with nucleus $N$ tangent to the line $\ell_\infty$ at $T$, and let…

Let $q$ be an even integer, and let $\mathrm{PG}(2,q)$ denote the Desarguesian projective plane of order $q$. A $(3q+4,4)$-arc is a point set of size $3q+4$ meeting every line in a…

Multi-arc smoothing lower-bound conjecture. The multi-arc $calpha$ has a smoothing that cannot be homotoped to a multi-arc with fewer than

A strong $(t\mod q)$-arc in a projective space is a point set with the corresponding arc multiplicity condition; an arc is lifted if it is obtained by the lifting construction from…

Arc-size conjecture. The largest size of an arc in $X$ over $F_q$ is $q+1$, except possibly when $q$ is even and $N\in\{3,q-1\}$, in which case the maximum value is $q+2$.

Let $v\ge 5$ and let a strong $(3\bmod 5)$-arc be a strong arc in $\operatorname{PG}(v-1,5)$ with the stated $(3\bmod 5)$ intersection property. Lifting conjecture. Every strong…

Let $2\le N\le q-2$. An $[n,n-N-1,N+2]_q$ MDS code is equivalent to an $n$-arc in $\mathrm{PG}(N,q)$. MDS conjecture. Every such code, or equivalently every such $n$-arc, has lengt…

Let $2\le N\le q-2$. In $\mathrm{PG}(N,q)$, a normal rational curve is a $(q+1)$-arc projectively equivalent to … A $(q+1)$-arc is complete if no further point can be added while p…

The conjecture on removing the modulo condition. The preceding statement holds without the modulo $\Psi[X]$; that is, the equality above holds as an equality of forms rather than o…

Let $q$ be a prime power and let $k$ satisfy … An arc in $\mathrm{PG}(k-1,q)$ is a set of points with no $k$ lying in a hyperplane. A normal rational curve is the projective set pa…

Let $q$ be a prime power and let $k$ satisfy … An arc in $\mathrm{PG}(k-1,q)$ is a set of points with no $k$ lying in a hyperplane. The MDS conjecture. Every arc in…

Let $q$ be an odd prime power, and let ${\mathcal{A}}$ be an $r$-arc in the projective plane ${\mathrm{PG}}(2,{\mathrm{GF}}(q))$, meaning a set of $r$ points with no three collinea…

Maximal-cardinality conjecture. The maximal cardinality of such a collection $\mathcal{A}$ on any connected, oriented surface with $\chi<0$ is

The $\sqrt{q}$ conjecture. The second largest complete arc in ${\mathrm{PG}}(2,q)$ has size

Let $N$ and $q$ be integers with $2\le N\le q-2$, and let $\mathrm{PG}(N,q)$ be the $N$-dimensional projective space over the finite field with $q$ elements. A normal rational curv…

Ball's conjecture. The matrix $\mathrm{P}_n$ has a vector of weight one in its column space for any arc that extends to a $(q+2)$-arc, when

Weight-one vector conjecture. The matrix $\mathrm{M}_{n}$ has a vector of weight one in its column space.

Let $PG(2,q)$ be the projective plane of order $q$, and let $t_{2}(2,q)$ denote the smallest size of a complete arc in $PG(2,q)$. The finite-range upper-bound conjecture. In…

Let $PG(2,q)$ be the projective plane of order $q$, and let $t_{2}(2,q)$ denote the smallest size of a complete arc in $PG(2,q)$. The logarithmic upper-bound conjecture. In…

Let $PG(2,q)$ be the projective plane of order $q$, let $\overline{t}_{2}(2,q)$ be the smallest known size of a complete arc, and let $M_q$ be the upper endpoint defined in the sou…

Let $t_{2}(2,q)$ denote the smallest size of a complete arc in the plane $PG(2,q)$. Upper-bound conjecture. It holds that … The preceding computations verify a stronger numerical b…

Let $q$ be even, and let a complete arc in the projective plane $PG(2,q)$ mean an arc that cannot be enlarged by adding another point. Two arcs are projectively equivalent when a p…