30 problems

Etzion–Silberstein conjecture. For every Ferrers diagram $\mathcal{F}$ of order $n$, every $1\le d\le n$, and every finite field $\mathbb{F}_q$, there exists an…

Ferrers diagram rank-metric code existence conjecture. For any $q,m,n,k$, there always exists a Ferrers diagram rank-metric code attaining this bound. This is important because cod…

Computational-effort conjecture. The computational effort for finite fields should be similar for all methods presented in the paper.

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…

Let $q,m,n,k$ be positive integers, with $q$ a prime power, and consider an $[n,k]_{\mathbb F_{q^m}/\mathbb F_q}$ code, meaning an $\mathbb F_{q^m}$-linear rank-metric code of leng…

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

Let $M$ range over the matrices considered in the paper, and let $m(r)$ be the rank-dependent quantity defined by … For $r=1,\dots,n+1$, consider the maximum of $\theta_M$ among ma…

Let $k<n-b1$, and let $\mathcal{C}_{F,k,A}=\Psi_A(\mathbb{F}_{q^m}\langle X^q\rangle_k)$ be a $q$CRT code, where $F=(f_1,\ldots,f_s)$ consists of pairwise suitably coprime lineariz…

Franch and Li's conjecture. For the general case,

Let the codes under consideration be special cases of Sheekey's constructed codes, and let the Galois group involved have different possible generators. Generator-dependence conjec…

Let $C\subseteq\mathbb{F}_{q^m}^N$ be any $[N,k]$ MRD code with generator matrix $G\in\mathbb{F}_{q^m}^{k\times N}$. Let $A\in\mathbb{F}_q^{N\times n}$ be a randomly sampled full-r…

Let $q$ be a prime power, let $u,d,r$ be positive integers, and let $Z$ be uniformly random in $\mathbb{F}_q^{u\times(d-1)r}$. Let $A\in\mathbb{F}_q^{(d-1)r\times(d-1)r}$ be constr…

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 $\alpha$ be as in the definition of $\mathcal{C}_{\rho}^{n,d}(\alpha)$, and let $\mathcal{M}^{n,d}$ be the set of $d$-element multisets with elements in…

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…

Attainability conjecture. The upper bound from Theorem, and the corresponding bound for Ferrers diagram rank-metric codes, can always be attained.

Let $\mathcal{F}$ be a Ferrers diagram, let $C_{\mathcal{F}}$ be an FDRM code with minimum rank distance $\delta$, and let $\mathcal{C}_{\mathcal{F}}$ be the corresponding lifted F…

Let $m$-MSR convolutional codes be constructed using superregular matrices of the type proposed in Almeida, Napp and Pinto (2013), Almeida and coauthors (2016), and Martínez-Peñas…

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 $\tilde{\ve{E}}$ be a random full-rank error matrix in the interleaved decoding setting considered above, and let the success probability be bounded using the proof of Loidreau…

Let $\mathcal A$ be sampled from the set of $\zeta$-dimensional subspaces of $\mathbb F_{q^m}^w$ that have a basis consisting of elements of full $\mathbb F_q$-rank. Equivalently,…

Let ${\mathcal{IG}(u;n,k)}$ be a $u$-interleaved Gabidulin code, and let $\mathbf z$ be an error pattern satisfying the rank condition in Lemma 1 (the interleaved-decoding failure…

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

Optimality conjecture. The bound given by Theorem is the optimal solution to this linear program.