8 problems

Let $n,d\in\mathbb N$ satisfy $1<d<n$, and let $\mathbb F$ be a finite field. An MRD code is a maximum rank-distance code. Puncturing-inclusion conjecture. There exists an…

Scatteredness conjecture. The polynomial $\psi_{m,h,s}$ is scattered if and only if $m$ and $h$ satisfy the assumptions in Theorem.

Let $\bm{M} \in \mathbb{F}_{q^m}^{k \times n}$ be a randomly chosen matrix with full $\mathbb{F}_{q^m}$-rank such that each block $\bm{M}^{(i)}$, for $i=1,\dots,\ell$, has full col…

Let $d53d$ be a field, let $S_{n,d}(d53d)$ denote the space of homogeneous degree-$d$ polynomials in $n$ variables, and let $\operatorname{d}_{\operatorname{ess}}$ be the essential…

Let $p$ be prime and let $n$ be a positive integer. Consider additive MRD codes in $M_{n\times n}(\mathbb{F}_p)$ with minimum distance $n$, where equivalence is taken under the usu…

Let a Gabidulin code be given together with a received word. A received word is a deep hole if its distance from the code equals the covering radius. NP-hardness conjecture. Decidi…

Norm-class conjecture. The set

Let ${\cal F}$ be a Ferrers diagram and let $\delta$ be a positive integer. For each $i$ with $0\leq i\leq\delta-1$, let $\nu_i$ be the number of dots in ${\cal F}$ outside the fir…