15 problems

MDS probability convergence conjecture. If

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 $\mathcal{E}$ 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 $q$ be a power of a prime, $(q-1)/2le n-k$, and $k<n\le q$. Let ${\bf a}=(\alpha_1,\ldots,\alpha_n)$ and let ${\bf w}$ denote the multiplier vector from the associated MDS code…

Let $2\le N\le q-2$. In $\mathrm{PG}(N,q)$, a normal rational curve is a $(q+1)$-arc projectively equivalent to … A $(q+1)$-arc is complete if no further point can be added while p…

Let $q=2^m$, with $m\equiv 3\pmod 4$ and $m\geq 5$, and define … Let ${\mathcal{C}}_{(f,q)}^{(2)}$ be the associated binary subfield code. Nine-weight conjecture. The code…

Let $q=2^m$, and let ${\mathcal{C}}_{(f,q)}^{(2)}$ be the binary subfield code defined by the trace representation for an oval polynomial $f$. Nine-weight conjecture. For odd…

Let $G=([n],E)$ be a $k$-uniform hypergraph, and let $q(G)$ be the smallest alphabet size for which there are encoding and decoding functions recovering every message from the symb…

Let $M$ be an $m\times n$ binary matrix satisfying the MDS condition. Let $N=\{\alpha_1,\dots,\alpha_n\}$ be independent indeterminates, let $V=V(N)$ be a generic $m\times n$ Vande…

Let $M$ be an $m\times n$ binary support matrix. For a nonempty set $I\subseteq\{1,\dots,m\}$, write $M_i$ for the $i$th row and $\operatorname{supp}(M_i)$ for its support. The mat…

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…

General covering-radius conjecture. For $2\le k\le q-2$,

Let $\mathbb{F}_q$ be a finite field, and let $[n,k]$ denote a linear code of length $n$ and dimension $k$ over $\mathbb{F}_q$. If $1<k<q$, then … except when $q$ is even and $k=3$…

Let $p$ be a prime, and let ${\mathbb C}$ be a code of length $p+1$ constructed from a generator matrix that is a conference matrix. Conference-matrix MDS conjecture. The code…