3 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…

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 $n$ and $d_{\text{R}}$ be positive integers, and let an $(n,n-d_{\text{R}}+1,d_{\text{R}})$ linear MRD code over $\mathrm{GF}(q^m)$ be given. Consider the subcode formed by its…