12 problems

Cyclic subspace code existence conjecture. There exists a cyclic subspace code

Existence conjecture. There exists a cyclic subspace code $\mathcal{C}\subseteq \mathcal{G}_q(n,k)$ such that

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

Let $\mathcal{U}$ be a linear code in the projective space $\mathbb{P}_q(n)$. An indecomposable basis is a basis of the vector space over $\mathbb{F}_2$ formed by the code whose el…

Let $\mathbb{P}_q(n)$ be the projective space of subspaces of $\mathbb{F}_q^n$, equipped with the notion of linear code introduced by Braun, Etzion, and Vardy. The BEV bound states…

Upper-bound conjecture. The quantity $A_2(2k,2k-2;k)$ is much smaller than

RRP bound conjecture. For any packet length $v\geq 7$ with $v\notin\{8,10\}$, the largest subspace codes that can be obtained by solving the RRP exceed the LMRD code bound and thus…

Expurgation-augmentation conjecture. Based on computational data, the largest subspace codes obtained by the expurgation-augmentation method exceed the LMRD code bound for all suff…

A parallelism in the Grassmannian $\mathcal{G}_{q}(n,k)$ is a packing of spreads that partitions $\mathcal{G}_{q}(n,k)$; a spread partitions the points of $\mathcal{G}_{q}(n,k)$ in…

Etzion's extension conjecture. The lower bound

In scalar linear network coding, coding coefficients are chosen from a finite field, whereas in vector network coding over a field of size $q$ and dimension $t$, each coefficient i…

Let $\mathcal{G}_{q}(n,k)$ denote the Grassmannian of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, and let a cyclic code be a subspace code invariant under multiplication by…