8 problems

A covering array $\operatorname{CA}(N,k,v)$ is an array with $N$ rows, $k$ columns, and entries from an alphabet of size $v$, such that every choice of two columns contains every o…

Covering-array nonexistence conjecture. A $CA(12,8,3)$ does not exist; consequently, …

Optimal covering-array conjecture. A $CA(n,QI(n,k),k)$ is an optimal covering array; equivalently, for positive integers $n,k$ with $n\geq k^2$, …

For a positive integer $n$, let $3\text{-}QI(n,2)$ denote the $3$-uniform qualitative independence hypergraph associated with binary covering arrays, and let $CAN(H,2)$ be the mini…

Let $v$ and $t$ be positive integers with $v\geqslant t$. Write $[v]=\{0,\dots,v-1\}$, let $\mathcal{S}_v$ be the set of permutations of $[v]$, and let $\mathcal{S}_{v,t}$ be the s…

Let \text{mathsf{CAN}}_\lambda(t,k,v) denote the smallest number of rows in a covering array with strength $t$, $k$ columns, alphabet size $v$, and covering multiplicity…

Levenshtein's conjecture. This property should hold for every $t\geqslant 3$: namely, the relevant boundary case $g(t+1,t)=1$, or equivalently, a PSCA$(t+1,t,1)$ should exist.

For integers $n$ and $k$, let $g(n,k)$ be the minimum covering multiplicity for a perfect sequence covering array with parameters $(n,k)$. The previously known exceptional values i…