14 problems

Algebraic intersection criterion. The arrangement $\lambda K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exists a unimodular transformation…

Let $P\subseteq \mathbb{R}^d$ be a non-hollow lattice polytope. A full-dimensional lattice polytope is called terminal if it is a direct sum of translates of terminal simplices, wh…

Let a finite collection of lattice vectors spanning $\mathbb{R}^d$ be cosimple if it has a linear dependence whose coefficients are all non-zero and have pairwise different absolut…

Let $Z_{\mathbf v}$ be a Strong Lonely Runner Zonotope, meaning a Lonely Runner Zonotope associated with a velocity vector $\mathbf v=(v_1,\dotsc,v_n)\in\mathbb{Z}_{>0}^n$ whose en…

Let $q$ be a prime power and let $\operatorname{PRS}(k)$ denote the projective Reed–Solomon code of dimension $k$ and length $q+1$. For $2\le k\le q-2$, let…

Covering conjecture. The covering radius satisfies

Let $G$ be an undirected, connected multigraph with genus $g\geq 1$ and $n\geq 2$ vertices. Let $\mathcal{N}_G$, ${\rm Crit}_{\triangle}(L_G)$ and…

Let $S=\operatorname{conv}(\{v_0,\ldots,v_d\})$ be a $d$-simplex with $\mathbf{0}\in S\setminus\{v_0,\ldots,v_d\}$ and with rational vertex directions. For each $i$, let…

Let $k,d\in\mathbb{N}$ be nonnegative integers with $k\geq2$. For every lattice $d$-polytope $P$ with $k$ interior lattice points, the prescribed-interior-points conjecture. … with…

Let $S$ be a $d$-simplex with the origin in its interior and with rational vertex directions. Let $\operatorname{Vol}_{\mathbb{Z}^d}(S)$ be normalized volume and let … where…

Let $\ell_q(r,3)$ be the length function, let $\ell_q(r,3,5)$ be the $d$-length function, let $s_q(r-1,2)$ be the smallest size of a $2$-saturating set in $\mathrm{PG}(r-1,q)$, and…

Let $S=\{\boldsymbol{z}_1,\dotsc,\boldsymbol{z}_m\}\subseteq\mathbb{Z}^n$ and let $Z$ be the lattice zonotope generated by $S$. The set $S$ is in LGP (general linear position) when…

Let $\mathcal E(n)$ be the projected lattice points on $S^2$, and let $N_n$ denote their number. The covering radius $M(P_1,\dots,P_N)$ of a configuration on $S^2$ is the least…

Let $Q\subset \mathbb F_2^n$ be an $\mathbb F_2$-linear code of size $n^t$, where $t=\Theta(1)$. Its covering radius is the maximum, over $u\in\mathbb F_2^n$, of the minimum Hammin…