13 problems

Prime power conjecture. A set of $n-1$ MOLS of order $n$ exists if and only if $n$ is a prime power.

Let $n$) have distinct prime-power factorization … and let … Write $N(n)$ for the maximum number of mutually orthogonal Latin squares of order $n$. MacNeish's conjecture. The equal…

Mallows–Sloane conjecture. For all $t\geq 8$, a tight $\operatorname{OA}(N,n,2,t)$ exists if and only if $n=t+1$.

Generalized sequence conjecture. The proposed infinite family of sequences produces an infinite orthogonal array in which the number of symbols, trials, and factors is countably in…

For fixed positive integers $t$ and $q$, consider all ordered orthogonal arrays obtained by the RUNS construction over the corresponding alphabet, and count the $t$-sets of columns…

The conjecture. For each odd $m$, there exists a Hadamard matrix $4m$ which yields an orthogonal array $\operatorname{OA}(4m,8,2,2)$ satisfying the conditions of Theorem.

Let $m\geq 4$ be even. Consider binary codes of length $2^m$ with size $2^{2^m-2m}$ and dual distance $2^{m-1}-2^{m/2-1}$; equivalently, let $F^*(n,2,t)$ denote the minimum size of…

Let $G^{\rm LP}$ be the linear-programming relaxation permutation symmetry group of the orthogonal-array defining integer linear program, with alphabet size $s$ and $k$ factors. Le…

Let $G^{\rm LP}$ be the linear-programming relaxation permutation symmetry group of the orthogonal-array defining integer linear program, with alphabet size $s=2$, strength $t$, an…

A frequency square of type $(n;\lambda)$ is an $n\times n$ array in which each symbol occurs exactly $\lambda$ times in every row and every column. Two frequency squares are orthog…

Appa et al.'s full-dimensionality conjecture. If $P^{(k,s)}_{n;I}\neq\emptyset$, then

Unique-completion problem. Is completion unique in the relevant setting of orthogonal arrays or mutually orthogonal Latin squares?

Actuality conjecture. For any fixed $k$ and all sufficiently large $N$, all plausible $\tau$-parities are actual for $(k,N)$.