9 problems

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…

Let $S$ be a sufficiently long sequence, and let $\hat{S}(k)$ denote its Fourier coefficients. The sequence is considered in the setting where its entries generate Hamming balls of…

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 $\mathscr{G}_{n,m}$ be the code in the family indexed by $n$ and $m$, and let $\mathscr{G}_{n,m}^{\bot}$ be its dual. Denote their relevant covering multiplicities by…

Let $\mathscr{D}_3$, $\mathscr{D}_4$, and $\mathscr{D}_5$ be the infinite families of codes described in the paper, with parameters indexed by $m$. An $(R,\lambda)$-APMCF code is a…

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 $K_{2}^{\mathcal{R}}(2,3,3)$ denote the minimum size of a $3$-covering code in the NRT space $\mathbb{Z}_{2}^{6}$ with poset $[2\cdot 3]$. Covering-code conjecture. … The const…

Projection-and-folding conjecture. In $4\mathbb{Z}^d4$, for colorings satisfying certain periodicity properties, the projection and folding method will lead to weighted cycles with…