6 problems

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. For subsets $I_1,\ldots,I_t\subseteq[n]$, define their weight by … and write…

Let $C\geq1$ be real, let $t\geq3$ be a positive integer, and let $k$ be a positive integer. For subsets $I_1,\ldots,I_t\subseteq[n]$, write $\operatorname{wt}(I_j:j\in J)$ for the…

Let $L$ be a positive integer, let $C\geq1$ be a positive real, and let $R=\frac{k}{n}$ be the rate of an $[n,k]$ Reed–Solomon code. A code is list-decodable from radius $\rho$ wit…

Let $n$ be a positive integer, let $1\le k<n$, let $R=\frac{k}{n}$, and let $\varepsilon>0$ satisfy $\varepsilon n\in\mathbb{Z}$. A Reed–Solomon code is list-decodable from radius…

For a code of rate $R$ over a finite field, list-decodable from relative radius $r$ with list size $L$ means that every Hamming ball of radius $r n$ contains at most $L$ codewords.…

A Reed–Solomon code is obtained by evaluating polynomials of degree less than $k$ at $n$ distinct elements of a finite field. For a prime power $q$, let an $[n,k]$-RS code be defin…