14 problems

Let $d$ be fixed, and consider the degree-$d$ Reed–Solomon code with evaluation set $FbbF_q$. A random puncturing is obtained by retaining a random subset of evaluation coordinates…

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 $q$ be a prime power, let $\theta$ be a primitive element when relevant, and let ${\rm RS}_k(\mathbb{F}_q)$ denote the full-length Reed–Solomon code of dimension $k$. For a vec…

Let $q$ be a prime power and let $\operatorname{PRS}(k)$ denote the projective Reed–Solomon code of dimension $k$ and length $q+1$. For $2\le k\le q-2$, let…

Reed–Solomon list-recoverability conjecture. The number of polynomials $P(X) \in \mathbb{F}_p[X]$ satisfying

Shangguan's full-rank conjecture. The matrix $M_{G,(J_1,\ldots,J_t)}$ has full column rank.

Let $[n,k]$ denote a Reed–Solomon code with an $\epsilon$-MSR repair scheme, meaning that the repair bandwidth for every single failed node is at most $(1+\epsilon)$ times the cut-…

List decodability of Reed-Solomon codes up to capacity. For every $\rho>0$, there is a constant $C_\rho$ such that every Reed-Solomon code of length $n$ and rate $\rho$ is list-dec…

Let $q$ be a prime power and let $m$ be an integer for which the stated MDS codes are defined. An $[q+1,m]_q$ Reed–Solomon code corresponds to a normal rational curve (RNC) in…

Let $q$ be a prime power and let $2\leq m\leq q-2$. An MDS extension by one digit of an $m$-dimensional Reed–Solomon code with evaluation set $GF(q)$ is an MDS code obtained by add…

Projective deep-hole conjecture. The set

Let $\mathbb{F}_q$ be a finite field, let $D=\mathbb{F}_q^*$ be the evaluation set of a primitive Reed–Solomon code with code length $n=q-1$, dimension $k$, and codebook…