4 problems

Fixed-rate puncture conjecture. For an $[n,k,d]_q$ GRS code ${\mathcal C}$ of fixed rate $R$, the average number of punctures needed for $\operatorname{GRS\text{-}cover}$ to succee…

Puncture-count conjecture. For an $[n,k,d]_q$ GRS code, the average number of punctures needed for $\operatorname{GRS\text{-}cover}$ to succeed in returning a codeword within the c…

Let $C$ be an $[n,k,d]_q$ generalized Reed–Solomon (GRS) code, and consider the covering algorithm that repeatedly punctures the received word and applies a GRS decoder. Let…

Let $\mathcal{S}_n$ be the set of permutations of $[n]=\{1,2,\dots,n\}$, with Hamming distance $d_H$. Define $f(n,s)$ to be the minimum size of a subset of $\mathcal{S}_n$ having c…