95 problems

A Reed–Solomon code is obtained by evaluating low-degree polynomials over a finite field at a chosen set of field elements; a random Reed–Solomon code uses a randomly selected eval…

Let $M(k,q)$ denote the maximal length of a non-trivial $q$-ary MDS code of dimension $k$. For $q>k$, the Main Conjecture on MDS codes. … This is a central parameter problem for MD…

Let $\operatorname{MEC}(k,q)$ denote the maximal length of a non-trivial $q$-ary MDS elliptic code of dimension $k$, where an elliptic code arises from an algebraic geometry code o…

Let $\mathcal S=(S_1,\dots,S_n)$ be a generic zero pattern for $k\times n$ matrices: for every $I\subseteq [k]$, it satisfies … An $(n,k)$-code attains $\mathcal S$ if it has a gen…

Alrabiah–Guruswami conjecture. This value of $N$ is exactly tight for high-rate MSR codes.

Let ${\bf F}_q$ be a finite field with $q$ elements, and let an MDS code over ${\bf F}_q$ be a linear code attaining the Singleton bound. MDS conjecture. The length of an MDS code…

Segre's main conjecture.

A classical $[n,k]_q$ MDS code is nontrivial when it is not one of the cases excluded by the stated bound. MDS conjecture. If there is a nontrivial $[n,k]_q$ MDS code, then…

Let ${\mathbb C}$ be an $(n,k,d)$ MDS code over ${\mathbb F}_{q}$. Its stopping redundancy is the minimum number of rows in a parity-check matrix of ${\mathbb C}$ whose associated…

GM-MDS conjecture. The MDS condition is sufficient, as well as necessary, to guarantee the existence of an MDS generator matrix over any field of size linear in the length of the c…

Let $C$ be a nontrivial linear $[m,n,d]_q$ MDS code over $\mathbb{F}_q$, where the exceptional cases with even $q$ and $n\in\{3,q-1\}$ are treated separately. The length $m$ is the…

Let $\Omega$ be an alphabet of size $q$. A complete $2$-recursive code is determined by a function $f:\Omega^2\to\Omega$, with successive symbols generated from the preceding two s…

A complete $2$-recursive code of length $n$ is specified by a function $f$ through the recurrence … for $l>2$, and it is an MDS code when its associated groupoid is a recursively d…

Let $\mathbb{F}_q$ be a finite field, and let $\mathcal{C}\subseteq\mathbb{F}_q^n$ be a non-trivial MDS code of dimension $k$. MDS conjecture. If such a code exists, then … except…

MDS probability convergence conjecture. If

MDS conjecture. The code length satisfies $n\le q+1$, unless $q$ is even and $k\in\{3,q-1\}$, in which case $n\le q+2$.

MDS conjecture. If $C$ is a nontrivial MDS $[n,k,d]_q$ code, then

Let $C$ be an $[n,k,d]$ MDS code over a field of order $q$, so that $d=n-k+1$, and suppose $d\geq 3$. MDS conjecture. The length $n$ must satisfy … except when $q=2^h$ and…

Maturana–Rashmi's conjecture. Under the Uniform Cost Assumption, every stable linear MDS $(n^{I},k^{I}={\lambda}^{F}k^{F};n^{F},k^{F})$ convertible code with $r^{F}<r^{I}$ and…

Let $\mathcal{C}$ be a nontrivial $[n,k,d]_q$ linear maximum distance separable (MDS) code, so that $d=n-k+1$ over $\mathbb{F}_q$. The parameters $n$, $k$, and $q$ denote its lengt…

Let an MDS code be a maximum-distance-separable code over the finite field $\mathbb{F}_q$, with length equal to the number of its coordinates. MDS conjecture. The length of an MDS…

Let $\mathbb{F}_q$ be a finite field of order $q$, with $q$ a prime power. An $[n,k,d]_q$ code is a linear code of length $n$, dimension $k$, and minimum distance $d$; it is MDS if…

Let $q$ be the size of the alphabet, and let $k$ denote the dimension of an MDS code of length $n$ and distance $d$. A code is non-trivial when $d>2$. MDS conjecture. No non-trivia…

Let $\mathcal{E}$ be an elliptic curve over $\mathbb{F}_q$, and let $C_L(D,G)$ be an $[n,k]$ MDS code from $\mathcal{E}$. Assume that $3\leq k\leq n-3$. Han and Ren's conjecture. I…

Let $C$ be an $[n,k]$ MDS elliptic code over $F_q$, where $n$ is its length. Li's conjecture. For any $epsilon>0$, there exists a constant $C_{epsilon}$ such that, whenever…