4 problems

Let $q_{i,j}=q_{i,j}^{[m,n]}$ be the non-negative integers obtained in Algorithm; these integers are independent of $m$ and $n$. For each $i\geq 1$, consider the sequence…

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.

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…

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